Solved: A student asked, “Since electrical potential is

A point charge is held stationary at the origin. A second point charge moves from the point to the point How much work is done by the electric force on

A point charge is held stationary at the origin. A second charge is placed at point and the electric potential energy of the pair of charges is When the second charge is moved to point the electric force on the charge does of work. What is the electric potential energy of the pair of charges when the second charge is at point

Energy of the Nucleus. How much work is needed to assemble an atomic nucleus containing three protons (such as Be) if we model it as an equilateral triangle of side 2.00 * 10-15 m with a proton at each vertex? Assume the protons started from very far away.

(a) How much work would it take to push two protons very slowly from a separation of (a typical atomic distance) to (a typical nuclear distance)? (b) If the protons are both released from rest at the closer distance in part (a), how fast are they moving when they reach their original separation?

A small metal sphere, carrying a net charge of is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of and mass 1.50 g, is projected toward When the two spheres are apart, is moving toward with speed (Fig. E23.5). Assume that the two spheres can be treated as point charges. You can ignore the force of gravity. (a) What is the speed of when the spheres are apart? (b) How close does get to

BIO Energy of DNA Base Pairing, I. (See Exercise 21.23.) (a) Calculate the electric potential energy of the adeninethymine bond, using the same combinations of molecules ( and ) as in Exercise 21.23. (b) Compare this energy with the potential energy of the protonelectron pair in the hydrogen atom.

BIO Energy of DNA Base Pairing, II. (See Exercise 21.24.) Calculate the electric potential energy of the guanine cytosine bond, using the same combinations of molecules ( , , and ) as in Exercise 21.24.

Three equal \(1.20-\mu \mathrm{C}\) point charges are placed at the corners of an equilateral triangle whose sides are 0.500 m long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Two protons are released from rest when they are 0.750 nm apart. (a) What is the maximum speed they will reach? When does this speed occur? (b) What is the maximum acceleration they will achieve? When does this acceleration occur?

Four electrons are located at the corners of a square on a side, with an alpha particle at its midpoint. How much work is needed to move the alpha particle to the midpoint of one of the sides of the square?

Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides d. Two of the point charges are identical and have charge q. If zero net work is required to place the three charges at the corners of the triangle, what must the value of the third charge be?

Starting from a separation of several meters, two protons are aimed directly toward each other by a cyclotron accelerator with speeds of measured relative to the earth. Find the maximum electrical force that these protons will exert on each other

A small particle has charge and mass It moves from point where the electric potential is to point where the electric potential is The electric force is the only force acting on the particle. The particle has speed at point What is its speed at point Is it moving faster or slower at than at Explain.

A particle with a charge of +4.20 nC is in a uniform electric field \(\vec E\) directed to the left. It is released from rest and moves to the left; after it has moved its kinetic energy is found to be \(+1.50 \times 10^{-6} J\).(a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of \(\vec E\)? Text Transcription: vec E +1.50 x 10^-6

A charge of is placed in a uniform electric field that is directed vertically upward and has a magnitude of What work is done by the electric force when the charge moves (a) to the right; (b) upward; (c) at an angle of downward from the horizontal?

Two stationary point charges \(+3.00\mathrm{\ nC}\) and \(+2.00\mathrm{\ nC}\) are separated by a distance of 50.0 cm. An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is 10.0 cm from the \(+3.00\mathrm{\ nC}\) charge?

Point charges and are placed at adjacent corners of a square for which the length of each side is 3.00 cm. Point a is at the center of the square, and point b is at the empty corner closest to . Take the electric potential to be zero at a distance far from both charges. (a) What is the electric potential at point a due to and ? (b) What is the electric potential at point b? (c) A point charge moves from point a to point b. How much work is done on by the electric forces exerted by and ? Is this work positive or negative?

Two charges of equal magnitude are held a distance apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). (b) Repeat part (a) for two charges having opposite signs.

Two point charges and are apart. Point is midway between them; point is from and from (Fig. E23.19). Take the electric potential to be zero at infinity. Find (a) the potential at point (b) the potential at point (c) the work done by the electric field on a charge of that travels from point to point

A positive charge is located at the point and a negative charge is located at the point (a) Derive an expression for the potential at points on the -axis as a function of the coordinate Take to be zero at an infinite distance from the charges. (b) Graph at points on the -axis as a function of over the range from to (c) Show that for the potential at a point on the positive -axis is given by (d) What are the answers to parts (a) and (c) if the two charges are interchanged so that is at and is at

A positive charge is fixed at the point and a negative charge is fixed at the point (a) Show the positions of the charges in a diagram. (b) Derive an expression for the potential at points on the -axis as a function of the coordinate Take to be zero at an infinite distance from the charges. (c) At which positions on the -axis is (d) Graph at points on the -axis as a function of in the range from to (e) What does the answer to part (b) become when Explain why this result is obtained.

Consider the arrangement of point charges described in Exercise 23.21. (a) Derive an expression for the potential at points on the -axis as a function of the coordinate Take to be zero at an infinite distance from the charges. (b) At which positions on the -axis is (c) Graph at points on the -axis as a function of in the range from to (d) What does the answer to part (a) become when Explain why this result is obtained.

(a) An electron is to be accelerated from to Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from to a halt?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are and respectively. (Take the potential to be zero at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the point charge?

Auniform electric field has magnitude and is directed in the negative -direction. The potential difference between point a (at x = 0.60 m) and point (at x = 0.90 m) is 240 V. (a) Which point, a or b is at the higher potential? (b) Calculate the value of E. (c) A negative point charge \(q = -0.200 \mu C\) is moved from b to a. Calculate the work done on the point charge by the electric field. Text Transcription: q = -0.200 muC

For each of the following arrangements of two point charges, find all the points along the line passing through both charges for which the electric potential V is zero (take V = 0 infinitely far from the charges) and for which the electric field E is zero: (a) charges +Q and +2Q separated by a distance d and (b) charges -Q and +2Q separated by a distance d (c) Are both V and E zero at the same places? Explain.

A thin spherical shell with radius is concentric with a larger thin spherical shell with radius . Both shells are made of insulating material. The smaller shell has charge distributed uniformly over its surface, and the larger shell has charge distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells. (a) What is the electric potential due to the two shells at the following distance from their common center: (i) ; (ii) ; (iii) ? (b) What is the magnitude of the potential difference between the surfaces of the two shells? Which shell is at higher potential: the inner shell or the outer shell?

A total electric charge of is distributed uniformly over the surface of a metal sphere with a radius of If the potential is zero at a point at infinity, find the value of the potential at the following distances from the center of the sphere: (a) (b) (c)

A uniformly charged, thin ring has radius and total charge An electron is placed on the rings axis a distance from the center of the ring and is constrained to stay on the axis of the ring. The electron is then released from rest. (a) Describe the subsequent motion of the electron. (b) Find the speed of the electron when it reaches the center of the ring

An infinitely long line of charge has linear charge density A proton (mass charge ) is from the line and moving directly toward the line at (a) Calculate the protons initial kinetic energy. (b) How close does the proton get to the line of charge?

A very long wire carries a uniform linear charge density Using a voltmeter to measure potential difference, you find that when one probe of the meter is placed from the wire and the other probe is farther from the wire, the meter reads (a) What is (b) If you now place one probe at from the wire and the other probe farther away, will the voltmeter read If not, will it read more or less than Why? (c) If you place both probes from the wire but from each other, what will the voltmeter read?

A very long insulating cylinder of charge of radius carries a uniform linear density of If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads

A very long insulating cylindrical shell of radius carries charge of linear density spread uniformly over its outer surface. What would a voltmeter read if it were connected between (a) the surface of the cylinder and a point above the surface, and (b) the surface and a point from the central axis of the cylinder?

A ring of diameter is fixed in place and carries a charge of uniformly spread over its circumference. (a) How much work does it take to move a tiny charged ball of mass from very far away to the center of the ring? (b) Is it necessary to take a path along the axis of the ring? Why? (c) If the ball is slightly displaced from the center of the ring, what will it do and what is the maximum speed it will reach?

A very small sphere with positive charge \(q=+8.00\ \mu\mathrm{C}\) is released from rest at a point 1.50 cm from a very long line of uniform linear charge density \(\lambda=+3.00\ \mu\mathrm{C}/\mathrm{m}\). What is the kinetic energy of the sphere when it is 4.50 cm from the line of charge if the only force on it is the force exerted by the line of charge?

Charge \(Q=5.00\ \mu\mathrm{C}\) is distributed uniformly over the volume of an insulating sphere that has radius R = 12.0 cm. A small sphere with charge \(q = +3.00\ \mu\mathrm{C}\) and mass \(6.00 \times 10^{-5}\mathrm{\ kg}\) is projected toward the center of the large sphere from an initial large distance. The large sphere is held at a fixed position and the small sphere can be treated as a point charge. What minimum speed must the small sphere have in order to come within 8.00 cm of the surface of the large sphere? Text Transcription: Q = 5.00 mu C q = +3.00 mu C 6.00 times 10^-5 kg

Axons. Neurons are the basic units of the nervous system. They contain long tubular structures called axons that propagate electrical signals away from the ends of the neurons. The axon contains a solution of potassium \(\left(\mathrm{K}^{+}\right)\) ions and large negative organic ions. The axon membrane prevents the large ions from leaking out, but the smaller \(\mathrm{K}^{+}\) ions are able to penetrate the membrane to some degree (Fig. E23.37). This leaves an excess negative charge on the inner surface of the axon membrane and an excess positive charge on the outer surface, resulting in a potential difference across the membrane that prevents further \(\mathrm{K}^{+}\) ions from leaking out. Measurements show that this potential difference is typically about 70 mV. The thickness of the axon membrane itself varies from about 5 to 10 nm, so we’ll use an average of 7.5 nm. We can model the membrane as a large sheet having equal and opposite charge densities on its faces. (a) Find the electric field inside the axon membrane, assuming (not too realistically) that it is filled with air. Which way does it point: into or out of the axon? (b) Which is at a higher potential: the inside surface or the outside surface of the axon membrane?

Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by 2.20 cm. (a) If the surface charge density for each plate has magnitude \(47.0\mathrm{\ nC}/\mathrm{m}^2\), what is the magnitude of \(\vec{E}\) in the region between the plates? (b) What is the potential difference between the two plates? (c) If the separation between the plates is doubled while the surface charge density is kept constant at the value in part (a), what happens to the magnitude of the electric field and to the potential difference?

Two large, parallel, metal plates carry opposite charges of equal magnitude. They are separated by 45.0 mm, and the potential difference between them is 360 V. (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge + 2.40 nC? c) Use the results of part (b) to compute the work done by the field on the particle as it moves from the higher-potential plate to the lower. (d) Compare the result of part (c) to the change of potential energy of the same charge, computed from the electric potential.

Electrical Sensitivity of Sharks. Certain sharks can detect an electric field as weak as To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5-V AA battery across these plates, how far apart would the plates have to be?

(a) Show that for a spherical shell of radius that has charge distributed uniformly over its surface, is the same as for a solid conductor with radius and charge (b) You rub an inflated balloon on the carpet and it acquires a potential that is lower than its potential before it became charged. If the charge is uniformly distributed over the surface of the balloon and if the radius of the balloon is what is the net charge on the balloon? (c) In light of its potential difference relative to you, do you think this balloon is dangerous? Explain.

(a) How much excess charge must be placed on a copper sphere in diameter so that the potential of its center, relative to infinity, is (b) What is the potential of the spheres surface relative to infinity?

The electric field at the surface of a charged, solid, copper sphere with radius 0.200 m is 3800 N/C directed toward the center of the sphere. What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere?

A very large plastic sheet carries a uniform charge density of on one face. (a) As you move away from the sheet along a line perpendicular to it, does the potential increase or decrease? How do you know, without doing any calculations? Does your answer depend on where you choose the reference point for potential? (b) Find the spacing between equipotential surfaces that differ from each other by What type of surfaces are these?

In a certain region of space, the electric potential is where and are positive constants. (a) Calculate the and -components of the electric field. (b) At which points is the electric field equal to zero?

In a certain region of space the electric potential is given by where V/m3 and 8.00 V/m3 . Calculate the magnitude and direction of the electric field at the point in the region that has coordinates m, y = 0.400 m, and z = 0.

A metal sphere with radius \(r_a\) is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius \(r_b\). There is charge +q on the inner sphere and charge -q on the outer spherical shell. (a) Calculate the potential V(r) for (i) \(r<r_{a}\); (ii) \(r_{a}<r<r_{b}\); (iii) \(r>r_{b}\). (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take V to be zero when r is infinite. (b) Show that the potential of the inner sphere with respect to the outer is \(V_{a b}=\frac{q}{4 \pi \epsilon_{0}}\left(\frac{1}{r_{a}}-\frac{1}{r_{b}}\right)\) (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude \(E(r)=\frac{V_{a b}}{\left(1 / r_{a}-1 / r_{b}\right)} \frac{1}{r^{2}}\) (d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance r from the center, where \(r>r_{b}\). (e) Suppose the charge on the outer sphere is not -q but a negative charge of different magnitude, say -Q. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.

A metal sphere with radius \(r_{a}=1.20 \mathrm{~cm}\) is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius \(r_{b}=9.60 \mathrm{~cm}\). Charge +q is put on the inner sphere and charge -q on the outer spherical shell. The magnitude of q is chosen to make the potential difference between the spheres 500 V, with the inner sphere at higher potential. (a) Use the result of Exercise 23.47(b) to calculate q. (b) With the help of the result of Exercise 23.47(a), sketch the equipotential surfaces that correspond to 500, 400, 300, 200, 100, and (c) In your sketch, show the electric field lines. Are the electric field lines and equipotential surfaces mutually perpendicular? Are the equipotential surfaces closer together when the magnitude of \(\overrightarrow{\boldsymbol{E}}\) is largest?

A very long cylinder of radius carries a uniform charge density of (a) Describe the shape of the equipotential surfaces for this cylinder. (b) Taking the reference level for the zero of potential to be the surface of the cylinder, find the radius of equipotential surfaces having potentials of (c) Are the equipotential surfaces equally spaced? If not, do they get closer together or farther apart as increases?

P A point charge is held fixed in space. From a horizontal distance of 6.00 cm, a small sphere with mass and charge is fired toward the fixed charge with an initial speed of 40.0 m/s. Gravity can be neglected. What is the acceleration of the sphere at the instant when its speed is 25.0 m/s?

A point charge \(q_1=4.00\mathrm{\ nC}\) is placed at the origin, and a second point charge \(q_2=-3.00\mathrm{\ nC}\) is placed on the x-axis at x = +20.0 cm. A third point charge \(q_3=2.00\mathrm{\ nC}\) is to be placed on the x-axis between \(q_{1}\) and \(q_{2}\). (Take as zero the potential energy of the three charges when they are infinitely far apart.) (a) What is the potential energy of the system of the three charges if \(q_{3}\) is placed at x = +10.0 cm? (b) Where should be \(q_{3}\) placed to make the potential energy of the system equal to zero?

A small sphere with mass and charge is released from rest a distance of 0.400 m above a large horizontal insulating sheet of charge that has uniform surface charge density . Using energy methods, calculate the speed of the sphere when it is 0.100 m above the sheet of charge?

Determining the Size of the Nucleus. When radium- 226 decays radioactively, it emits an alpha particle (the nucleus of helium), and the end product is radon-222. We can model this decay by thinking of the radium-226 as consisting of an alpha particle emitted from the surface of the spherically symmetric radon- 222 nucleus, and we can treat the alpha particle as a point charge. The energy of the alpha particle has been measured in the laboratory and has been found to be 4.79 MeV when the alpha particle is essentially infinitely far from the nucleus. Since radon is much heavier than the alpha particle, we can assume that there is no appreciable recoil of the radon after the decay. The radon nucleus contains 86 protons, while the alpha particle has 2 protons and the radium nucleus has 88 protons. (a) What was the electric potential energy of the alpharadon combination just before the decay, in MeV and in joules? (b) Use your result from part (a) to calculate the radius of the radon nucleus.

A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha particle (a helium nucleus) has essentially four times the mass and two times the charge of a proton. Find the maximum speed and maximum acceleration of each of these particles. When do these maxima occur: just following the release of the particles or after a very long time?

A particle with charge is in a uniform electric field directed to the left. Another force, in addition to the electric force, acts on the particle so that when it is released from rest, it moves to the right. After it has moved the additional force has done of work and the particle has of kinetic energy. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of the electric field?

In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electrons speed. (b) Obtain an expression for the electrons kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using Give your numerical result in joules and in electron volts

CALC A vacuum tube diode consists of concentric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is not a linear function of the position, even with planar geometry, but is given by where is the distance from the cathode and is a constant, characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is and the potential difference between electrodes is (a) Determine the value of (b) Obtain a formula for the electric field between the electrodes as a function of (c) Determine the force on an electron when the electron is halfway between the electrodes.

Two oppositely charged, identical insulating spheres, each in diameter and carrying a uniform charge of magnitude are placed apart center to center (Fig. P23.58). (a) If a voltmeter is connected between the nearest points ( and ) on their surfaces, what will it read? (b) Which point, or is at the higher potential? How can you know this without any calculations?

An Ionic Crystal. Figure P23.59 shows eight point charges arranged at the corners of a cube with sides of length The values of the charges are and as shown. This is a model of one cell of a cubic ionic crystal. In sodium chloride (NaCl), for instance, the positive ions are and the negative ions are (a) Calculate the potential energy of this arrangement. (Take as zero the potential energy of the eight charges when they are infinitely far apart.) (b) In part (a), you should have found that Explain the relationship between this result and the observation that such ionic crystals exist in nature.

(a) Calculate the potential energy of a system of two small spheres, one carrying a charge of and the other a charge of with their centers separated by a distance of Assume zero potential energy when the charges are infinitely separated. (b) Suppose that one of the spheres is held in place and the other sphere, which has a mass of is shot away from it. What minimum initial speed would the moving sphere need in order to escape completely from the attraction of the fixed sphere? (To escape, the moving sphere would have to reach a velocity of zero when it was infinitely distant from the fixed sphere.)

The Ion. The ion is composed of two protons, each of charge and an electron of charge and mass The separation between the protons is The protons and the electron may be treated as point charges. (a) Suppose the electron is located at the point midway between the two protons. What is the potential energy of the interaction between the electron and the two protons? (Do not include the potential energy due to the interaction between the two protons.) (b) Suppose the electron in part (a) has a velocity of magnitude in a direction along the perpendicular bisector of the line connecting the two protons. How far from the point midway between the two protons can the electron move? Because the masses of the protons are much greater than the electron mass, the motions of the protons are very slow and can be ignored. (Note: A realistic description of the electron motion requires the use of quantum mechanics, not Newtonian mechanics.)

A small sphere with mass hangs by a thread between two parallel vertical plates apart (Fig. P23.62). The plates are insulating and have uniform surface charge densities and The charge on the sphere is What potential difference between the plates will cause the thread to assume an angle of with the vertical?

Coaxial Cylinders. A long metal cylinder with radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is \(\lambda\), and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential V(r) for (i) r < a; (ii) a < r < b; (iii) r > b. (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r = b. (b) Show that the potential of the inner cylinder with respect to the outer is \(V_{a b}=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{b}{a}\) (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude \(E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r}\) (d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it (Fig. P23.64). A large potential difference is established between the wire and the outer cylinder, with the wire at higher potential; this sets up a strong electric field directed radially outward. When ionizing radiation enters the device, it ionizes a few air molecules. The free electrons produced are accelerated by the electric field toward the wire and, on the way there, ionize many more air molecules. Thus a current pulse is produced that can be detected by appropriate electronic circuitry and converted to an audible click. Suppose the radius of the central wire is and the radius of the hollow cylinder is What potential difference between the wire and the cylinder produces an electric field of at a distance of from the axis of the wire? (The wire and cylinder are both very long in comparison to their radii, so the results of Problem 23.63 apply.)

CP Deflection in a CRT. Cathode-ray tubes (CRTs) are often found in oscilloscopes and computer monitors. In Fig. P23.65 an electron with an initial speed of is projected along the axis midway between the deflection plates of a cathoderay tube. The potential difference between the two plates is 22.0 V and the lower plate is the one at higher potential. (a) What is the force (magnitude and direction) on the electron when it is between the plates? (b) What is the acceleration of the electron (magnitude and direction) when acted on by the force in part (a)? (c) How far below the axis has the electron moved when it reaches the end of the plates? (d) At what angle with the axis is it moving as it leaves the plates? (e) How far below the axis will it strike the fluorescent screen S?

CP Deflecting Plates of an Oscilloscope. The vertical deflecting plates of a typical classroom oscilloscope are a pair of parallel square metal plates carrying equal but opposite charges. Typical dimensions are about on a side, with a separation of about The potential difference between the plates is 25.0 V. The plates are close enough that we can ignore fringing at the ends. Under these conditions: (a) how much charge is on each plate, and (b) how strong is the electric field between the plates? (c) If an electron is ejected at rest from the negative plate, how fast is it moving when it reaches the positive plate?

Electrostatic precipitators use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants. One form of precipitator consists of a vertical, hollow, metal cylinder with a thin wire, insulated from the cylinder, running along its axis (Fig. P23.67). A large potential difference is established between the wire and the outer cylinder, with the wire at lower potential. This sets up a strong radial electric field directed inward. The field produces a region of ionized air near the wire. Smoke enters the precipitator at the bottom, ash and dust in it pick up electrons, and the charged pollutants are accelerated toward the outer cylinder wall by the electric field. Suppose the radius of the central wire is the radius of the cylinder is and a potential difference of is established between the wire and the cylinder. Also assume that the wire and cylinder are both very long in comparison to the cylinder radius, so the results of Problem 23.63 apply. (a) What is the magnitude of the electric field midway between the wire and the cylinder wall? (b) What magnitude of charge must a ash particle have if the electric field computed in part (a) is to exert a force ten times the weight of the particle?

A disk with radius R has uniform surface charge density \(\sigma\). (a) By regarding the disk as a series of thin concentric rings, calculate the electric potential V at a point on the disk’s axis a distance x from the center of the disk. Assume that the potential is zero at infinity. (Hint: Use the result of Example 23.11 in Section 23.3.) (b) Calculate \(-\partial V / \partial x\). Show that the result agrees with the expression for \(E_{x}\) calculated in Example 21.11 (Section 21.5).

(a) From the expression for obtained in Problem 22.42, find the expressions for the electric potential as a function of both inside and outside the cylinder. Let at the surface of the cylinder. In each case, express your result in terms of the charge per unit length of the charge distribution. (b) Graph and as functions of from to

A thin insulating rod is bent into a semicircular arc of radius a and a total electric charge Q is distributed uniformly along the rod. Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.

CALC Self-Energy of a Sphere of Charge. A solid sphere of radius contains a total charge distributed uniformly throughout its volume. Find the energy needed to assemble this charge by bringing infinitesimal charges from far away. This energy is called the self-energy of the charge distribution. (Hint: After you have assembled a charge in a sphere of radius how much energy would it take to add a spherical shell of thickness having charge Then integrate to get the total energy.)

(a) From the expression for obtained in Example 22.9 (Section 22.4), find the expression for the electric potential V as a function of r both inside and outside the uniformly charged sphere. Assume that V = 0 at infinity. (b) Graph V and E as functions of from r = 0 to r = 3R.

Charge is distributed uniformly over the volume of an insulating sphere that has radius . What is the potential difference between the center of the sphere and the surface of the sphere?

An insulating spherical shell with inner radius and outer radius 25.0 cm and outer radius 60.0 cm carries a charge of \(+150.0 \mu \mathrm{C}\) uniformly distributed over its outer surface (see Exercise 23.41). Point a is at the center of the shell, point is on the inner surface, and point is on the outer surface. (a) What will a voltmeter read if it is connected between the following points: (i) a and b; (ii) b and c; (iii) c and infinity; (iv) a and c? (b) Which is at higher potential: (i) a or b; (ii) b or c; (iii) a or c? (c) Which, if any, of the answers would change sign if the charge were \(-150 \mu \mathrm{C}\)?

Exercise 23.41 shows that, outside a spherical shell with uniform surface charge, the potential is the same as if all the charge were concentrated into a point charge at the center of the sphere. (a) Use this result to show that for two uniformly charged insulating shells, the force they exert on each other and their mutual electrical energy are the same as if all the charge were concentrated at their centers. (Hint: See Section 13.6.) (b) Does this same result hold for solid insulating spheres, with charge distributed uniformly throughout their volume? (c) Does this same result hold for the force between two charged conducting shells? Between two charged solid conductors? Explain.

Two plastic spheres, each carrying charge uniformly distributed throughout its interior, are initially placed in contact and then released. One sphere is 60.0 cm in diameter, has mass 50.0 g, and contains \(-10.0\ \mu \mathrm{C}\) of charge. The other sphere is 40.0 cm in diameter, has mass 150.0 g, and contains \(-30.0\ \mu \mathrm{C}\) of charge. Find the maximum acceleration and the maximum speed achieved by each sphere (relative to the fixed point of their initial location in space), assuming that no other forces are acting on them. (Hint: The uniformly distributed charges behave as though they were concentrated at the centers of the two spheres.) Text Transcription: -10.0 mu C -30.0 mu C

Use the electric field calculated in Problem 22.45 to calculate the potential difference between the solid conducting sphere and the thin insulating shell

Consider a solid conducting sphere inside a hollow conducting sphere, with radii and charges specified in Problem 22.44. Take as Use the electric field calculated in Problem 22.44 to calculate the potential at the following values of (a) (at the outer surface of the hollow sphere); (b) (at the inner surface of the hollow sphere); (c) (at the surface of the solid sphere); (d) (at the center of the solid sphere).

Electric charge is distributed uniformly along a thin rod of length a, with total charge Q. Take the potential to be zero at infinity. Find the potential at the following points (Fig. P23.79): (a) point P, a distance x to the right of the rod, and (b) point R, a distance y above the right-hand end of the rod. (c) In parts (a) and (b), what does your result reduce to as x or y becomes much larger than a?

(a) If a spherical raindrop of radius carries a charge of uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infi- nite distance from the raindrop.) (b) Two identical raindrops, each with radius and charge specified in part (a), collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

Two metal spheres of different sizes are charged such that the electric potential is the same at the surface of each. Sphere has a radius three times that of sphere Let and be the charges on the two spheres, and let and be the electric-field magnitudes at the surfaces of the two spheres. What are (a) the ratio and (b) the ratio

An alpha particle with kinetic energy 11.0 MeV makes a head-on collision with a lead nucleus at rest. What is the distance of the closest approach of the two particles? (Assume that the lead nucleus remains stationary and that it may be treated as a point charge. The atomic number of lead is 82. The alpha particle is a helium nucleus, with atomic number 2.)

A metal sphere with radius \(R_1\) has a charge \(Q_1\). Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius \(R_2\) that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere; (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.

Use the charge distribution and electric field calculated in Problem 22.65. (a) Show that for \(r\ge R\) the potential is identical to that produced by a point charge Q (Take the potential to be zero at infinity.) (b) Obtain an expression for the electric potential valid in the region \(r\le R\).

Nuclear Fusion in the Sun. The source of the sun's energy is a sequence of nuclear reactions that occur in its core. The first of these reactions involves the collision of two protons, which fuse together to form a heavier nucleus and release energy. For this process, called nuclear fusion, to occur, the two protons must first approach until their surfaces are essentially in contact. (a) Assume both protons are moving with the same speed and they collide head-on. If the radius of the proton is \(1.2 \times 10^{-15}\) m, what is the minimum speed that will allow fusion to occur? The charge distribution within a proton is spherically symmetric, so the electric field and potential outside a proton are the same as if it were a point charge. The mass of the proton is \(1.67 \times 10^{-27}\) kg. (b) Another nuclear fusion reaction that occurs in the sun's core involves a collision between two helium nuclei, each of which has 2.99 times the mass of the proton, charge +2e and radius \(1.7 \times 10^{-15}\) m. Assuming the same collision geometry as in part (a), what minimum speed is required for this fusion reaction to take place if the nuclei must approach a center-to-center distance of about \(3.5 \times 10^{-15}\) m? As for the proton, the charge of the helium nucleus is uniformly distributed throughout its volume. (c) In Section 18.3 it was shown that the average translational kinetic energy of a particle with mass m in a gas at absolute temperature T is \(\frac{3}{2} k T\), where k, is the Boltzmann constant (given in Appendix F). For two protons with kinetic energy equal to this average value to be able to undergo the process described in part (a), what absolute temperature is required? What absolute temperature is required for two average helium nuclei to be able to undergo the process described in part (b)? (At these temperatures, atoms are completely ionized, so nuclei and electrons move separately.) (d) The temperature in the sun's core is about \(1.5 \times 10^{7}\) K. How does this compare to the temperatures calculated in part (c)? How can the reactions described in parts (a) and (b) occur at all in the interior of the sun? (Hint: See the discussion of the distribution of molecular speeds in Section 18.5.)

The electric potential in a region of space is given by where is a constant. (a) Derive an expression for the electric field at any point in this region. (b) The work done by the field when a test charge moves from the point to the origin is measured to be Determine (c) Determine the electric field at the point (0, 0, 0.250 m). (d) Show that in every plane parallel to the -plane the equipotential contours are circles. (e) What is the radius of the equipotential contour corresponding to and

Nuclear Fission. The unstable nucleus of uranium236 can be regarded as a uniformly charged sphere of charge Q=+92e and radius \(R=7.4\times10^{-15}\) .In nuclear fission, this can divide into two smaller nuclei, each with half the charge and half the volume of the original uranium-236 nucleus. This is one of the reactions that occurred in the nuclear weapon that exploded over Hiroshima, Japan, in August 1945. (a) Find the radii of the two “daughter” nuclei of charge +46e (b) In a simple model for the fission process, immediately after the uranium-236 nucleus has undergone fission, the “daughter” nuclei are at rest and just touching, as shown in Fig. P23.87. Calculate the kinetic energy that each of the “daughter” nuclei will have when they are very far apart. (c) In this model the sum of the kinetic energies of the two “daughter” nuclei, calculated in part (b), is the energy released by the fission of one uranium236 nucleus. Calculate the energy released by the fission of 10.0 kg of uranium-236. The atomic mass of uranium-236 is 236 u where 1 u = 1 atomic mass unit \(=1.66\times10^{-24}\) kg. Express your answer both in joules and in kilotons of TNT (1 kiloton of TNT releases \(4.18\times10^{12}\) J when it explodes). (d) In terms of this model, discuss why an atomic bomb could just as well be called an “electric bomb.”

In a certain region, a charge distribution exists that is spherically symmetric but nonuniform. That is, the volume charge density depends on the distance from the center of the distribution but not on the spherical polar angles and The electric potential due to this charge distribution is where is a constant having units of and is a constant having units of meters. (a) Derive expressions for for the regions and [Hint: Use Eq. (23.23).] Explain why has only a radial component. (b) Derive an expression for in each of the two regions and [Hint: Use Gausss law for two spherical shells, one of radius and the other of radius The charge contained in the infinitesimal spherical shell of radius is ] (c) Show that the net charge contained in the volume of a sphere of radius greater than or equal to is zero. [Hint: Integrate the expressions derived in part (b) for over a spherical volume of radius greater than or equal to ] Is this result consistent with the electric field for that you calculated in part (a)?

In experiments in which atomic nuclei collide, head-on collisions like that described in Problem 23.82 do happen, but near misses are more common. Suppose the alpha particle in Problem 23.82 was not aimed at the center of the lead nucleus, but had an initial nonzero angular momentum (with respect to the stationary lead nucleus) of magnitude where is the magnitude of the initial momentum of the alpha particle and What is the distance of closest approach? Repeat for and

A hollow, thin-walled insulating cylinder of radius R and length L (like the cardboard tube in a roll of toilet paper) has charge Q uniformly distributed over its surface. (a) Calculate the electric potential at all points along the axis of the tube. Take the origin to be at the center of the tube, and take the potential to be zero at infinity. (b) Show that if \(L\ \ll\ R\), the result of part (a) reduces to the potential on the axis of a ring of charge of radius R. (See Example 23.11 in Section 23.3.) (c) Use the result of part (a) to find the electric field at all points along the axis of the tube.

The charge of an electron was first measured by the American physicist Robert Millikan during 19091913. In his experiment, oil is sprayed in very fine drops (around in diameter) into the space between two parallel horizontal plates separated by a distance A potential difference is maintained between the parallel plates, causing a downward electric field between them. Some of the oil drops acquire a negative charge because of frictional effects or because of ionization of the surrounding air by x rays or radioactivity. The drops are observed through a microscope. (a) Show that an oil drop of radius at rest between the plates will remain at rest if the magnitude of its charge is q = 4p 3 rr 3 gd VAB where is the density of the oil. (Ignore the buoyant force of the air.) By adjusting to keep a given drop at rest, the charge on that drop can be determined, provided its radius is known. (b) Millikans oil drops were much too small to measure their radii directly. Instead, Millikan determined by cutting off the electric field and measuring the terminal speed of the drop as it fell. (We discussed the concept of terminal speed in Section 5.3.) The viscous force on a sphere of radius moving with speed through a fluid with viscosity is given by Stokess law: When the drop is falling at the viscous force just balances the weight of the drop. Show that the magnitude of the charge on the drop is Within the limits of their experimental error, every one of the thousands of drops that Millikan and his coworkers measured had a charge equal to some small integer multiple of a basic charge That is, they found drops with charges of and so on, but none with values such as or A drop with charge has acquired one extra electron; if its charge is it has acquired two extra electrons, and so on. (c) A charged oil drop in a Millikan oil-drop apparatus is observed to fall at constant speed in if The same drop can be held at rest between two plates separated by if How many excess electrons has the drop acquired, and what is the radius of the drop? The viscosity of air is and the density of the oil is

Two point charges are moving to the right along the x-axis. Point charge 1 has charge \(q_1=2.00\ \mu\mathrm{C}\), mass \(m_1 = 6.00 \times 10^{-5}\mathrm{\ kg}\), and speed \(v_{1}\). Point charge 2 is to the right of \(q_{1}\) and has charge \(q_2 = -5.00\ \mu\mathrm{C}\), mass \(m_2 = 3.00 \times 10^{-5}\mathrm{\ kg}\), and speed \(v_{2}\). At a particular instant, the charges are separated by a distance of and have speeds \(v_1 = 400\ \mathrm{m}/\mathrm{s}\quad\text{ and } v_2 = 1300 \mathrm{\ m}/\mathrm{s}\). The only forces on the particles are the forces they exert on each other. (a) Determine the speed \(v_{\mathrm{cm}}\) of the center of mass of the system. (b) The relative energy \(E_{\mathrm{rel}}\) of the system is defined as the total energy minus the kinetic energy contributed by the motion of the center of mass: \(E_{\mathrm{rel}}=E-\frac{1}{2}\left(m_{1}+m_{2}\right) v_{\mathrm{cm}}^{2}\) where \(E=\frac{1}{2} m_{1} v_{1}^{2}+\frac{1}{2} m_{2} v_{2}^{2}+q_{1} q_{2} / 4 \pi \epsilon_{0} r\) is the total energy of the system and r is the distance between the charges. Show that \(E_{\text {rel }}=\frac{1}{2} \mu v^{2}+q_{1} q_{2} / 4 \pi \epsilon_{0} r, \text { where } \mu=m_{1} m_{2} /\left(m_{1}+m_{2}\right)\) is called the reduced mass of the system and \(v=v_{2}-v_{1}\) is the relative speed of the moving particles. (c) For the numerical values given above, calculate the numerical value of \(E_{\mathrm{rel}}\). (d) Based on the result of part (c), for the conditions given above, will the particles escape from one another? Explain. (e) If the particles do escape, what will be their final relative speed when \(r \rightarrow \infty\)? If the particles do not escape, what will be their distance of maximum separation? That is, what will be the value of r when v = 0? Repeat parts (c)–(e) for \(v_1 = 400 \mathrm{\ m}/\mathrm{s} \text { and } v_2 = 1800 \mathrm{\ m}/\mathrm{s}\) when the separation is 9.00 mm. Text Transcription: q_1=2.00 mu C m_1 = 6.00 times 10^-5 kg v_1 q_1 q_2 = -5.00 mu C m_2 = 3.00 times 10^-5 kg v_2 v_1 = 400 m/s and v_2 = 1300 m/s v_cm E_rel E_rel = E-1/2(m_1+m_2)v_cm^2 E = 1/2 m_1v_1^2 + 1/2 m_2v_2^2 + q_1q_2/4 pi epsilon_0 r E_rel = 1/2 mu v^2 + q_1q_2/4 pi epsilon_0 r, where mu = m_1m_2/(m_1 + m_2) v = v_2 - v_1 r rightarrow infty v_1 = 400 m/s and v_2 = 1800 m/s

Problem 1DQ A student asked, “Since electrical potential is always proportional to potential energy, why bother with the concept of potential at all?” How would you respond?

Problem 3E Energy of the Nucleus. How much work is needed to assemble an atomic nucleus containing three protons (such as Li) if we model it as an equilateral triangle of side 2.00 X 10-15 m with a proton at each vertex? Assume the protons started from very far away.

Problem 4DQ Since potential can have any value you want depending on the choice of the reference level of zero potential, how does a voltmeter know what to read when you connect it between two points?

Problem 4E (a) How much work would it take to push two protons very slowly from a separation of 2.00 X 10-10 m (a typical atomic distance) to 3.00 X 10-15 m (a typical nuclear distance)? (b) If the protons are both released from rest at the closer distance in part (a), how fast are they moving when they reach their original separation?

Problem 7DQ If you carry out the integral of the electric field \(\int \vec{E} \cdot \overrightarrow{d l}\) for a closed path like that shown in Fig. Q23.7, the integral will always be equal to zero, independent of the shape of the path and independent of where charges may be located relative to the path. Explain why. Equation Transcription: Text Transcription: Integral vector E.vector dl Vector dl Vector E

Problem 8DQ The potential difference between the two terminals of an AA battery (used in flashlights and portable stereos) is 1.5 V. If two AA batteries are placed end to end with the positive terminal of one battery touching the negative terminal of the other, what is the potential difference between the terminals at the exposed ends of the combination? What if the two positive terminals are touching each other? Explain your reasoning.

Problem 8E Three equal 1.20-?C point charges are placed at the corners of an equilateral triangle whose sides are 0.500 m long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Problem 9DQ It is easy to produce a potential difference of several thousand volts between your body and the floor by scuffing your shoes across a nylon carpet. When you touch a metal doorknob, you get a mild shock. Yet contact with a power line of comparable voltage would probably be fatal. Why is there a difference?

Problem 9E Two protons are released from rest when they are 0.750 nm apart. (a) What is the maximum speed they will reach? When does this speed occur? (b) What is the maximum acceleration they will achieve? When does this acceleration occur?

Problem 10DQ If the electric potential at a single point is known, can \(\vec{E}\) at that point be determined? If so, how? If not, why not? Equation Transcription: Text Transcription: Vector E

Problem 10E Four electrons are located at the corners of a square 10.0 nm on a side, with an alpha particle at its midpoint. How much work is needed to move the alpha particle to the midpoint of one of the sides of the square?

Problem 11DQ Because electric field lines and equipotential surfaces are always perpendicular, two equipotential surfaces can never cross; if they did, the direction of \(\vec{E}\) would be ambiguous at the crossing points. Yet two equipotential surfaces appear to cross at the center of Fig. 23.23c. Explain why there is no ambiguity about the direction of \(\vec{E}\) in this particular case. Equation Transcription: Text Transcription: Vector E Vector E

Problem 11E Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides d. Two of the point charges are identical and have charge q. If zero net work is required to place the three charges at the corners of the triangle, what must the value of the third charge be?

Problem 12DQ A uniform electric field is directed due east. Point B is 2.00 m west of point A, point C is 2.00 m east of point A, and point D is 2.00 m south of A. For each point, B, C, and D, is the potential at that point larger, smaller, or the same as at point A? Give the reasoning behind your answers.

Problem 12E Starting from a separation of several meters, two protons are aimed directly toward each other by a cyclotron accelerator with speeds of 1000 km/s, measured relative to the earth. Find the maximum electrical force that these protons will exert on each other.

Problem 13DQ We often say that if point A is at a higher potential than point B, A is at positive potential and B is at negative potential. Does it necessarily follow that a point at positive potential is positively charged, or that a point at negative potential is negatively charged? Illustrate your answers with clear, simple examples.

Problem 13E A small particle has charge -5.00 µC and mass 2.00 X 10-4 kg. It moves from point A, where the electric potential is VA = +200 V, to point B, where the electric potential is VB = +800 V. The electric force is the only force acting on the particle. The particle has speed 5.00 m/s at point A. What is its speed at point B? Is it moving faster or slower at B than at A? Explain.

Problem 14DQ A conducting sphere is to be charged by bringing in positive charge a little at a time until the total charge is Q. The total work required for this process is alleged to be proportional to Q2. Is this correct? Why or why not?

Problem 14E A particle with a charge of \(+4.20\) nC is in a uniform electric field \(\vec{E}\) directed to the left. It is released from rest and moves to the left; after it has moved 6.00 cm, its kinetic energy is found to be \(+1.50 \times 10^{-6}\) J. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of \(\vec{E}\)? Equation Transcription: Text Transcription: +4.20 Vector E +1.50x10^-6 Vector E

Problem 15DQ Three pairs of parallel metal plates (A, B, and C) are connected as shown in Fig. Q23.15, and a battery maintains a potential of 1.5 V across ab. What can you say about the potential difference across each pair of plates? Why?

A charge of 28.0 nC is placed in a uniform electric field that is directed vertically upward and has a magnitude of \(4.00 \times 10^4\mathrm{\ V}/\mathrm{m}\). What work is done by the electric force when the charge moves (a) 0.450 m to the right; (b) 0.670 m upward; (c) 2.60 m at an angle of \(45.0^{\circ}\) downward from the horizontal?

Problem 16DQ A conducting sphere is placed between two charged parallel plates such as those shown in Fig. 23.2. Does the electric field inside the sphere depend on precisely where between the plates the sphere is placed? What about the electric potential inside the sphere? Do the answers to these questions depend on whether or not there is a net charge on the sphere? Explain your reasoning.

Problem 19E Two point charges \(q_{1}=+2.40\) nC and \(q_{2}=-6.50\) nC are 0.100 m apart. Point A is midway between them; point B is 0.080 m from \(q_{1}\) and 0.060 m from \(q_{2}\) (Fig. E23.19). Take the electric potential to be zero at infinity. Find (a) the potential at point A; (b) the potential at point B; (c) the work done by the electric field on a charge of 2.50 nC that travels from point B to point A. . Equation Transcription: Text Transcription: q_1=+2.40 q_2=-6.50 q_1 q_2 q_1 q_2

A positive point charge is placed near a very large conducting plane. A professor of physics asserted that the field caused by this configuration is the same as would be obtained by removing the plane and placing a negative point charge of equal magnitude in the mirror-image position behind the initial position of the plane. Is this correct? Why or why not? (Hint: Inspect Fig. 23.23b.)

Problem 23E (a) An electron is to be accelerated from 3.00 X 106 m/s to 8.00 X 106 m/s. Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from 8.00 X 106 m/s to a halt?

Problem 24E At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 V and 16.2 V/m, respectively. (Take V = 0 at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the point charge?

A uniform electric field has magnitude E and is directed in the negative x-direction. The potential difference between point a (at x = 0.60 m) and point b (at x = 0.90 m) is 240 V. (a) Which point, a or b is at the higher potential? (b) Calculate the value of E. (c) A negative point charge \(q=-0.200 \mu \mathrm{C}\) is moved from b to a. Calculate the work done on the point charge by the electric field.

For each of the following arrangements of two point charges, find all the points along the line passing through both charges for which the electric potential V is zero (take V = 0 infinitely far from the charges) and for which the electric field E is zero: (a) charges +Q and +2Q separated by a distance d , and (b) charges -Q and +2Q separated by a distance d . (c) Are both V and E zero at the same places? Explain.

Problem 28E A total electric charge of 3.50 nC is distributed uniformly over the surface of a metal sphere with a radius of 24.0 cm. If the potential is zero at a point at infinity, find the value of the potential at the following distances from the center of the sphere: (a) 48.0 cm; (b) 24.0 cm; (c) 12.0 cm.

Problem 29E A uniformly charged, thin ring has radius 15.0 cm and total charge +24.0 nC. An electron is placed on the ring’s axis a distance 30.0 cm from the center of the ring and is constrained to stay on the axis of the ring. The electron is then released from rest. (a) Describe the subsequent motion of the electron. (b) Find the speed of the electron when it reaches the center of the ring.

Problem 30E An infinitely long line of charge has linear charge density 5.00 × 10?12 C/m. A proton (mass 1.67 × 10?27 kg, charge + 1.60 × 10?19 C) is 18.0 cm from the line and moving directly toward the line at 1.50 × 103 m/s.(a) Calculate the proton’s initial kinetic energy. (b) How close does the proton get to the line of charge?

A very long wire carries a uniform linear charge density \(\lambda\). Using a voltmeter to measure potential difference, you find that when one probe of the meter is placed 2.50 cm from the wire and the other probe is 1.00 cm farther from the wire, the meter reads 575 V. (a) What is \(\lambda\)? (b) If you now place one probe at 3.50 cm from the wire and the other probe 1.00 cm farther away, will the voltmeter read 575 V? If not, will it read more or less than 575 V? Why? (c) If you place both probes 3.50 cm from the wire but 17.0 cm from each other, what will the voltmeter read?

A very long insulating cylinder of charge of radius 2.50 cm carries a uniform linear density of 15.0 nC/m. If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads 175 V?

Problem 36E Charge Q = 5.00 µC is distributed uniformly over the volume of an insulating sphere that has radius R = 12.0 cm. A small sphere with charge q = +3.00 µC and mass 6.00 X 10-5 kg is projected toward the center of the large sphere from an initial large distance. The large sphere is held at a fixed position and the small sphere can be treated as a point charge. What minimum speed must the small sphere have in order to come within 8.00 cm of the surface of the large sphere?

Problem 37E Axons. Neurons are the basic units of the nervous system. They contain long tubular structures called axons that propagate electrical signals away from the ends of the neurons. The axon contains a solution of potassium \(\left(\mathrm{K}^{+}\right)\) ions and large negative organic ions. The axon membrane prevents the large ions from leaking out, but the smaller \(\mathrm{K}^{+}\) ions are able to penetrate the membrane to some degree (Fig. E23.37). This leaves an excess negative charge on the inner surface of the axon membrane and an excess positive charge on the outer surface, resulting in a potential difference across the membrane that prevents further \(\mathrm{K}^{+}\) ions from leaking out. Measurements show that this potential difference is typically about 70 mV. The thickness of the axon membrane itself varies from about 5 to 10 nm, so we’ll use an average of 7.5 nm. We can model the membrane as a large sheet having equal and opposite charge densities on its faces. (a) Find the electric field inside the axon membrane, assuming (not too realistically) that it is filled with air. Which way does it point: into or out of the axon? (b) Which is at a higher potential: the inside surface or the outside surface of the axon membrane? Equation Transcription: Text Transcription: (K^+) K^+ K^+ K^+ K^+ K^+ K^+ K^+

Problem 38E Two large, parallel conducting plates carrying opposite charges of equal magnitude are separated by 2.20 cm. (a) If the surface charge density for each plate has magnitude \(47.0 \mathrm{nC} / \mathrm{m}^{2}\), what is the magnitude of \(\vec{E}\) in the region between the plates? (b) What is the potential difference between the two plates? (c) If the separation between the plates is doubled while the surface charge density is kept constant at the value in part (a), what happens to the magnitude of the electric field and to the potential difference? Equation Transcription: Text Transcription: 47.0 nC/m^2 Vector E

Two large, parallel, metal plates carry opposite charges of equal magnitude. They are separated by 45.0 mm, and the potential difference between them is 360 V. (a) What is the magnitude of the electric field (assumed to be uniform) in the region between the plates? (b) What is the magnitude of the force this field exerts on a particle with charge +2.40 nC? (c) Use the results of part (b) to compute the work done by the field on the particle as it moves from the higher-potential plate to the lower. (d) Compare the result of part (c) to the change of potential energy of the same charge, computed from the electric potential.

Problem 40E BIO Electrical Sensitivity of Sharks. Certain sharks can detect an electric field as weak as 1.0 µV/m. To grasp how weak this field is, if you wanted to produce it between two parallel metal plates by connecting an ordinary 1.5-V AA battery across these plates, how far apart would the plates have to be?

Problem 41E (a) Show that V for a spherical shell of radius R, that has charge q distributed uniformly over its surface, is the same as V for a solid conductor with radius R and charge q. (b) You rub an inflated balloon on the carpet and it acquires a potential that is 1560 V lower than its potential before it became charged. If the charge is uniformly distributed over the surface of the balloon and if the radius of the balloon is 15 cm, what is the net charge on the balloon? (e) In light of its 1200-V potential difference relative to you, do you think this balloon is dangerous? Explain.

A particle with charge +7.60 nC is in a uniform electric field directed to the left. Another force, in addition to the electric force, acts on the particle so that when it is released from rest, it moves to the right. After it has moved 8.00 cm, the additional force has done \(6.50 \times 10^{-5}\mathrm{\ J}\) of work and the particle has \(4.35 \times 10^{-5}\mathrm{\ J}\) of kinetic energy. (a) What work was done by the electric force? (b) What is the potential of the starting point with respect to the end point? (c) What is the magnitude of the electric field? Text Transcription: 6.50 times 10^-5 J 4.35 times 10^-5 J

Problem 56P In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius r. Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron’s speed. (b) Obtain an expression for the electron’s kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using r = 5.29 × 10?11 m. Give your numerical result in joules and in electron volts.

Problem 57P CALC A vacuum tube diode consists of concentric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is given by V(x) = Cx4/3 where x is the distance from the cathode and C is a constant, characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is 13.0 mm and the potential difference between electrodes is 240 V. (a) Determine the value of C. (b) Obtain a formula for the electric field between the electrodes as a function of x. (c) Determine the force on an electron when the electron is halfway between the electrodes.

Problem 58P Two oppositely charged, identical insulating spheres, each 50.0 cm in diameter and carrying a uniform charge of magnitude 250 \(\mu \mathrm{C}\), are placed 1.00 m apart center to center (Fig. P23.58). (a) If a voltmeter is connected between the nearest points (a and b) on their surfaces, what will it read? (b) Which point, a or b is at the higher potential? How can you know this without any calculations? Equation Transcription: Text Transcription: uC

An Ionic Crystal. Figure P23.59 shows eight point charges arranged at the corners of a cube with sides of length d. The values of the charges are +q and -q, as shown. This is a model of one cell of a cubic ionic crystal. In sodium chloride (NaCl), for instance, the positive ions are \(Na^+\) and the negative ions are \(CI^-\). (a) Calculate the potential energy U of this arrangement. (Take as zero the potential energy of the eight charges when they are infinitely far apart.) (b) In part (a), you should have found that U < 0. Explain the relationship between this result and the observation that such ionic crystals exist in nature.

Problem 60P (a) Calculate the potential energy of a system of two small spheres, one carrying a charge of 2.00 µC and the other a charge of -3.50 µC, with their centers separated by a distance of 0.180 m. Assume that U = 0 when the charges are infinitely separated. (b) Suppose that one sphere is held in place; the other sphere, with mass 1.50 g, is shot away from it. What minimum initial speed would the moving sphere need to escape completely from the attraction of the fixed sphere? (To escape, the moving sphere would have to reach a velocity of zero when it is infinitely far from the fixed sphere.)

Problem 61P The \(H_{2}^{+}\) Ion. The \(H_{2}^{+}\) ion is composed of two protons, each of charge \(+e=1.60 \times 10^{-19}\) C, and an electron of charge \(-e\) and mass \(9.11 \times 10^{-31}\) kg. The separation between the protons is \(1.07 \times 10^{-10}\) m. The protons and the electron may be treated as point charges. (a) Suppose the electron is located at the point midway between the two protons. What is the potential energy of the interaction between the electron and the two protons? (Do not include the potential energy due to the interaction between the two protons.) (b) Suppose the electron in part (a) has a velocity of magnitude \(1.50 \times 10^{6} \mathrm{~m} / \mathrm{s}\) in a direction along the perpendicular bisector of the line connecting the two protons. How far from the point midway between the two protons can the electron move? Because the masses of the protons are much greater than the electron mass, the motions of the protons are very slow and can be ignored. (Note: A realistic description of the electron motion requires the use of quantum mechanics, not Newtonian mechanics.) Equation Transcription: Text Transcription: H_2^+ H_2^+ +e=1.60x10^-19 -e 9.11x10^-31 1.07x10^-10 1.50x^6 m/s

Problem 62P A small sphere with mass 1.50 g hangs by a thread between two parallel vertical plates 5.00 cm apart (Fig. P23.62). The plates are insulating and have uniform surface charge densities \(+\sigma\) and \(-\sigma\). The charge on the sphere is \(q=8.90 \times 10^{-6}\) C. What potential difference between the plates will cause the thread to assume an angle of \(30.0^{\circ}\) with the vertical? Equation Transcription: Text Transcription: +sigma -sigma q=8.90x10^-6 30.0deg 30.0deg

Coaxial Cylinders. A long metal cylinder with radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is \lambda, and there is an equal negative charge per unit length on the outer cylinder. (a) Calculate the potential \(V(r)\) for \(\text{ (i) }r\ <\ a\); \(\text{ (ii) }a\ <\ r\ <\ b\); \(\text{ (iii) }r\ >\ b\) (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take \(V=0\) at \(r=b\). (b) Show that the potential of the inner cylinder with respect to the outer is \(V_{a b}=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{b}{a}\) (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude \(E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r}\) (d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?

Problem 67P Electrostatic precipitators use electric forces to remove pollutant particles from smoke, in particular in the smokestacks of coal-burning power plants. One form of precipitator consists of a vertical, hollow, metal cylinder with a thin wire, insulated from the cylinder, running along its axis (Fig. P23.67). A large potential difference is established between the wire and the outer cylinder, with the wire at lower potential. This sets up a strong radial electric field directed inward. The field produces a region of ionized air near the wire. Smoke enters the precipitator at the bottom, ash and dust in it pick up electrons, and the charged pollutants are accelerated toward the outer cylinder wall by the electric field. Suppose the radius of the central wire is 90 \(\mu \mathrm{m}\), the radius of the cylinder is 14.0 cm, and a potential difference of 50.0 kV is established between the wire and the cylinder. Also assume that the wire and cylinder are both very long in comparison to the cylinder radius, so the results of Problem 23.63 apply. (a) What is the magnitude of the electric field midway between the wire and the cylinder wall? (b) What magnitude of charge must a 30.0-\(\mu \mathrm{g}\) ash particle have if the electric field computed in part (a) is to exert a force ten times the weight of the particle? Equation Transcription: kV Text Transcription: um ug kV

Problem 68P A disk with radius R has uniform surface charge density \(\sigma\). (a) By regarding the disk as a series of thin concentric rings, calculate the electric potential V at a point on the disk’s axis a distance from the center of the disk. Assume that the potential is zero at infinity. (Hint: Use the result of Example 23.11 in Section 23.3.) (b) Calculate \(-\partial V / \partial x\). Show that the result agrees with the expression for \(E_{x}\) calculated in Example 21.11 (Section 21.5). Equation Transcription: Text Transcription: Sigma -Partialdv/Partialdx E_x

Problem 69P (a) From the expression for E obtained in problem find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length ? of the charge distribution. (b) Graph V and E as functions of from r = 0 to r = 3R. Problem: A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume ?. (a) Derive the expression for the electric field inside the volume at a distance r from the axis of the cylinder in terms of the charge density ?. (b) What is the electric field at a point outside the volume in terms of the charge per unit length ? in the cylinder? (c) Compare the answers to parts (a) and (b) for r = R. (d) Graph the electric-field magnitude as a function of r from r = 0 to r = 3R.

Problem 70P CALC A thin insulating rod is bent into a semicircular arc of radius a, and a total electric charge Q is distributed uniformly along the rod. Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.

Problem 71P CALC Self-Energy of a Sphere of Charge. A solid sphere of radius R contains a total charge Q distributed uniformly throughout its volume. Find the energy needed to assemble this charge by bringing infinitesimal charges from far away. This energy is called the “self-energy” of the charge distribution. (Hint: After you have assembled a charge q in a sphere of radius r, how much energy would it take to add a spherical shell of thickness dr having charge dq? Then integrate to get the total energy.)

Problem 72P (a) From the expression for obtained in Example 22.9 (Section 22.4), find the expression for the electric potential as a function of both inside and outside the uniformly charged sphere. Assume that \(V=0\) at infinity. (b) Graph and as functions of from \(r=0\) to \(r=3 R\). Equation Transcription: Text Transcription: V=0 r=0 r=3R

Problem 85P Nuclear Fusion in the Sun. The source of the sun’s energy is a sequence of nuclear reactions that occur in its core. The first of these reactions involves the collision of two protons, which fuse together to form a heavier nucleus and release energy. For this process, called nuclear fusion, to occur, the two protons must first approach until their surfaces are essentially in contact. (a) Assume both protons are moving with the same speed and they collide head-on. If the radius of the proton is what is the minimum speed that will allow fusion to occur? The charge distribution within a proton is spherically symmetric, so the electric field and potential outside a proton are the same as if it were a point charge. The mass of the proton is (b) Another nuclear fusion reaction that occurs in the sun’s core involves a collision between two helium nuclei, each of which has 2.99 times the mass of the proton, charge and radius Assuming the same collision geometry as in part (a), what minimum speed is required for this fusion reaction to take place if the nuclei must approach a center-to-center distance of about As for the proton, the charge of the helium nucleus is uniformly distributed throughout its volume. (c) In Section 18.3 it was shown that the average translational kinetic energy of a particle with mass m in a gas at absolute temperature is where k is the Boltzmann constant (given in Appendix F). For two protons with kinetic energy equal to this average value to be able to undergo the process described in part (a), what absolute temperature is required? What absolute temperature is required for two average helium nuclei to be able to undergo the process described in part (b)? (At these temperatures, atoms are completely ionized, so nuclei and electrons move separately.) (d) The temperature in the sun’s core is about How does this compare to the temperatures calculated in part (c)? How can the reactions described in parts (a) and (b) occur at all in the interior of the sun? (Hint: See the discussion of the distribution of molecular speeds in Section 18.5.) Equation Transcription: Text Transcription: 1.2x10^-15 1.67x10^-27 +2e 1.7x10^-15 3.5x10^-15 3/2kT 1.5x10^7

Problem 86P The electric potential in a region of space is given by \(V(x, y, z)=A\left(x^{2}-3 y^{2}+z^{2}\right)\) Where is a constant. (a) Derive an expression for the electric field at any point in this region. (b) The work done by the field when a 1.50-\(\mu \mathrm{C}\) test charge moves from the point \((x, y, z)=(0,0,0.250 \mathrm{~m})\) to the origin is measured to be \(6.00 \times 10^{-5}\) J.Determine A. (c) Determine the electric field at the point (0, 0, 0.250 m). (d) Show that in every plane parallel to the \(x z\)-plane the equipotential contours are circles. (e) What is the radius of the equipotential contour corresponding to \(V=1280\) V and \(y=2.00\) m? Equation Transcription: Text Transcription: V(x,y,z)=A(x^2-3y^2+z^2) uC (x,y,z)=(0,0,0.250 m) 6.00x10^-5 xz V=1280 y=2.00

Problem 87P Nuclear Fission. The unstable nucleus of uranium 236 can be regarded as a uniformly charged sphere of charge \(Q=+92 e\) and radius \(R=7.4 \times 10^{-15}\) m. In nuclear fission, this can divide into two smaller nuclei, each with half the charge and half the volume of the original uranium-236 nucleus. This is one of the reactions that occurred in the nuclear weapon that exploded over Hiroshima, Japan, in August 1945. (a) Find the radii of the two “daughter” nuclei of charge \(+46 e\). (b) In a simple model for the fission process, immediately after the uranium-236 nucleus has undergone fission, the “daughter” nuclei are at rest and just touching, as shown in Fig. P23.87. Calculate the kinetic energy that each of the “daughter” nuclei will have when they are very far apart. (c) In this model the sum of the kinetic energies of the two “daughter” nuclei, calculated in part (b), is the energy released by the fission of one uranium-236 nucleus. Calculate the energy released by the fission of 10.0 kg of uranium-236. The atomic mass of uranium-236 is 236 u, where 1 \(u=1\) atomic mass unit \(=1.66 \times 10^{-24}\) kg. Express your answer both in joules and in kilotons of TNT (1 kiloton of TNT releases \(4.18 \times 10^{12}\) J when it explodes). (d) In terms of this model, discuss why an atomic bomb could just as well be called an “electric bomb.” Equation Transcription: Text Transcription: Q=+92e R=7.4x10^-15 +46e u=1 =1.66x10^-24 4.18x10^12 Q=+92e Q=+46e Q=+46e

Problem 88CP In a certain region, a charge distribution exists that is spherically symmetric but nonuniform. That is, the volume charge density \(\rho(r)\) depends on the distance from the center of the distribution but not on the spherical polar angles \(\theta\) and \(\Phi\). The electric potential \(V(r)\) due to this charge distribution is \(V(r)=\frac{\rho_{0} a^{2}}{18 \epsilon_{0}}\left[1-3\left(\frac{r}{a}\right)^{2}+2\left(\frac{r}{a}\right)^{3}\right] \text { for } r \leq a\) \(V(r)=0 \text { for } r \geq a\) Where \(\rho_{0}\) is a constant having units of \(\mathrm{C} / \mathrm{m}^{3}\) and a is a constant having units of meters. (a) Derive expressions for \(\vec{E}\) for the regions \(r \leq a\) and \(r \geq a\). [Hint: Use Eq. (23.23).] Explain why \(\vec{E}\) has only a radial component. (b) Derive an expression for \(\rho(r)\) in each of the two regions \(r \leq a\) and \(r \geq a\). [Hint: Use Gauss’s law for two spherical shells, one of radius and the other of radius \(r+d r\). The charge contained in the infinitesimal spherical shell of radius dr is \(d q=4 \pi r^{2} \rho(r) d r\).] (c) Show that the net charge contained in the volume of a sphere of radius greater than or equal to is zero. [Hint: Integrate the expressions derived in part (b) for \(\rho(r)\) over a spherical volume of radius greater than or equal to a. ] Is this result consistent with the electric field for \(r>a\) that you calculated in part (a)? Equation Transcription: Text Transcription: rho(r) Theta Phi V(r) V(r)=rho_0a^2/18E_0[1-3(r/a)^2+2(r/a)^3] for r <a V(r)=0 for r> a rho_0 C/m^3 vectorE r<a r>a vectorE r<a r>a r+dr dq=4pir^2rho(r) dr rho(r) r>a

Problem 89CP In experiments in which atomic nuclei collide, head-on collisions like that described in Problem do happen, but “near misses” are more common Suppose the alpha particle in Problem was not “aimed” at the center of the lead nucleus, but had an initial nonzero angular momentum (with respect to the stationary lead nucleus) of magnitude L = P0b. where p0 is the magnitude of the initial momentum of the alpha particle and b = 1.00 × 10?12 m. What is the distance of closest approach? Repeat for b = 1.00 × 10?13 m and b = 1.00 × 10?14 m. Problem: An alpha particle with kinetic energy 11.0 MeV makes a head-on collision with a lead nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the lead nucleus remains stationary and that it may be treated as a point charge. The atomic number of lead is 82. The alpha particle is a helium nucleus, with atomic number 2.)

The Millikan Oil-Drop Experiment. The charge of an electron was first measured by the American physicist Robert Millikan during 1909–1913. In his experiment, oil is sprayed in very fine drops (around \(10^{-4}mm\) in diameter) into the space between two parallel horizontal plates separated by a distance d. A potential difference \(V_{AB}\) is maintained between the parallel plates, causing a downward electric field between them. Some of the oil drops acquire a negative charge because of frictional effects or because of ionization of the surrounding air by x rays or radioactivity. The drops are observed through a microscope. (a) Show that an oil drop of radius at rest between the plates will remain at rest if the magnitude of its charge is \(q=\frac{4 \pi}{3} \frac{\rho r^{3} g d}{V_{A B}}\) Where \(\rho\) is the density of the oil. (Ignore the buoyant force of the air.) By adjusting \(V_{A B}) to keep a given drop at rest, the charge on that drop can be determined, provided its radius is known. (b) Millikan’s oil drops were much too small to measure their radii directly. Instead, Millikan determined by cutting off the electric field and measuring the terminal speed \(v_t\) of the drop as it fell. (We discussed the concept of terminal speed in Section 5.3.) The viscous force on a sphere of radius moving with speed through a fluid with viscosity \(\eta\) is given by Stokes’s law: \(F=6 \pi \eta r v\). When the drop is falling at \(v_t\), the viscous force just balances the weight ww = mg of the drop. Show that the magnitude of the charge on the drop is \(q=18 \pi \frac{d}{V_{A B}} \sqrt{\frac{\eta^{3} v_{\mathrm{t}}^{3}}{2 \rho g}}\) Within the limits of their experimental error, every one of the thousands of drops that Millikan and his coworkers measured had a charge equal to some small integer multiple of a basic charge e. That is, they found drops with charges of \(\pm 2 e, \pm 5 e\), and so on, but none with values such as 0.76e or 2.49e. A drop with charge -e has acquired one extra electron; if its charge is -2e, it has acquired two extra electrons, and so on. (c) A charged oil drop in a Millikan oil-drop apparatus is observed to fall 1.00 mm at constant speed in 39.3 s if \(V_{A B}=0\). The same drop can be held at rest between two plates separated by 1.00 mm if \(V_{A B}=9.16 V\). How many excess electrons has the drop acquired, and what is the radius of the drop? The viscosity of air is \(1.81 \times 10^{-5} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\), and the density of the oil is \(824 \mathrm{~kg} / \mathrm{m}^{3}\).

Problem 92CP Two point charges are moving to the right along the \(x \text {-axis }\). Point charge 1 has charge \(q_{1}=2.00 \mu C\), mass \(m_{1}=6.00 \times 10^{-5} \mathrm{~kg}\), and speed \(v_{1}\). Point charge 2 is to the right of \(q_{1}\) and has charge \(q_{2}=-5.00 \mu C\), mass \(m_{2}=3.00 \times 10^{-5} \mathrm{~kg}\), and speed \(v_{2}\). At a particular instant, the charges are separated by a distance of 9.00 mm and have speeds \(v_{1}=400 \mathrm{~m} / \mathrm{s}\) and \(v_{2}=1300 \mathrm{~m} / \mathrm{s}\). The only forces on the particles are the forces they exert on each other. (a) Determine the speed \(v_{c m}\) of the center of mass of the system. (b) The relative energy \(E_{r e l}\) of the system is defined as the total energy minus the kinetic energy contributed by the motion of the center of mass: \(E_{\mathrm{rel}}=E-\frac{1}{2}\left(m_{1}+m_{2}\right) v_{\mathrm{cm}}^{2}\) Where \(E=\frac{1}{2} m_{1} v_{1}^{2}+\frac{1}{2} m_{2} v_{2}^{2}+q_{1} q_{2} / 4 \pi \epsilon_{0} r\) is the total energy of the system and is the distance between the charges. Show that \(E_{r e l}=\frac{1}{2} \mu v^{2}+q_{1} q_{2} / \pi \epsilon_{0} r\), where \(\mu=m_{1} m_{2} /\left(m_{1}+m_{2}\right)\) is called the reduced mass of the system and \(v=v_{2}-v_{1}\) is the relative speed of the moving particles. (c) For the numerical values given above, calculate the numerical value of (d) Based on the result of part (c), for the conditions given above, will the particles escape from one another? Explain. (e) If the particles do escape, what will be their final relative speed when \(r \rightarrow \infty\)? If the particles do not escape, what will be their distance of maximum separation? That is, what will be the value of when \(v=0\)? (f) Repeat parts (c)–(e) for \(v_{1}=400 \mathrm{~m} / \mathrm{s}\) and \(v_{2}=1800 \mathrm{~m} / \mathrm{s}\) when the separation is 9.00 mm. Equation Transcription: Text Transcription: x-axis q_1=2.00 uC m_1=6.00x10^-5 kg v_1 q_1 q_2=-5.00 uC m_2=3.00x10^-5 kg v_2 v_1=400 m/s v_2=1300 m/s E_rel E_rel=E-1/2(m_1+m_2)v_cm^2 E-1/2m_1v_1^2+1/2m_2v_2^2+q_1q_2/4piE_0r E_rel=1/2uv^2+q_1q_2/4piE_0r u=m_1m_2/(m_1+m_2) v=v_2-v_1 E_rel r->infinity v=0 v_1=400 m/s v_2=1800 m/s

Problem 1E A point charge q1 = +2.40 µC is held stationary at the origin. A second point charge q2 = -4.30 µC moves from the point x = 0.150 m, y = 0 to the point x = 0.250 m, y = 0.250 m. How much work is done by the electric force on q2?

The potential (relative to a point at infinity) midway between two charges of equal magnitude and opposite sign is zero. Is it possible to bring a test charge from infinity to this midpoint in such a way that no work is done in any part of the displacement? If so, describe how it can be done. If it is not possible, explain why.

A point charge \(q_{1}\) is held stationary at the origin. A second charge \(q_{2}\) is placed at point a, and the electric potential energy of the pair of charges is \(+5.\times 10^{-8} J\). When the second charge is moved to point b, the electric force on the charge does \(-1.9 \times 10^{-8} J\) of work. What is the electric potential energy of the pair of charges when the second charge is at point b? Text Transcription: q_1 q_2 +5.4 x 10^-8 J -1.9 x 10^-8 J

Problem 3DQ Is it possible to have an arrangement of two point charges separated by a finite distance such that the electric potential energy of the arrangement is the same as if the two charges were infinitely far apart? Why or why not? What if there are three charges? Explain.

If \(\bar{E}\) is zero everywhere along a certain path that leads from point to point , what is the potential difference between those two points? Does this mean that \(\bar{E}\) is zero everywhere along any path from to Explain. Equation transcription: Text transcription: bar{E}

A small metal sphere, carrying a net charge of \(q_{1}=-2.80 \mu C\), is held in a stationary position by insulating supports. A second small metal sphere, with a net charge of \(q_{2}=-7.80 \mu C\) and mass , is projected toward \(q_{1}\). When the two spheres are apart, \(q_{2}\) is moving toward \(q_{1}\) with speed (Fig. E23.5). Assume that the two spheres can be treated as point charges. You can ignore the force of gravity. (a) What is the speed of \(q_{2}\) when the spheres are apart? (b) How close does \(q_{2}\) get to \(q_{1}\) ? Equation transcription: Text transcription: q{1}=-2.80 \mu C q{2}=-7.80 \mu C q{2} q{1}

If \(\mathbf{\vec{E}}\) is zero throughout a certain region of space, is the potential necessarily also zero in this region? Why or why not? If not, what can be said about the potential?

Problem 16E Two stationary point charges +3.00 nC and +2.00 nC are separated by a distance of 50.0 cm. An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is 10.0 cm from the +3.00-nC charge?

Problem 17DQ A conductor that carries a net charge Q has a hollow, empty cavity in its interior. Does the potential vary from point to point within the material of the conductor? What about within the cavity? How does the potential inside the cavity compare to the potential within the material of the conductor?

Point charges \(q_1=+2.00\ \mu\mathrm{C}\text{ and }q_2=-2.00\ \mu\mathrm{C}\) are placed at adjacent corners of a square for which the length of each side is 3.00 cm. Point a is at the center of the square, and point b is at the empty corner closest to \(q_{2}\). Take the electric potential to be zero at a distance far from both charges. (a) What is the electric potential at point a due to \(q_{1}\) and \(q_{2}\)? (b) What is the electric potential at point b? (c) A point charge \(q_3=-5.00\ \mu\mathrm{C}\) moves from point a to point b. How much work is done on \(q_{3}\) by the electric forces exerted by \(q_{1}\) and \(q_{2}\)? Is this work positive or negative?

Problem 18DQ A high-voltage dc power line falls on a car, so the entire metal body of the car is at a potential of 10,000 V with respect to the ground. What happens to the occupants (a) when they are sitting in the car and (b) when they step out of the car? Explain your reasoning.

Problem 18E Two point charges of equal magnitude Q are held a distance d apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). (b) Repeat part (a) for two point charges having opposite signs.

When a thunderstorm is approaching, sailors at sea sometimes observe a phenomenon called “St. Elmo’s fire,” a bluish flickering light at the tips of masts. What causes this? Why does it occur at the tips of masts? Why is the effect most pronounced when the masts are wet? (Hint: Seawater is a good conductor of electricity.)

Auniform electric field has magnitude and is directed in the negative -direction. The potential difference between point a (at x = 0.60 m) and point (at x = 0.90 m) is 240 V. (a) Which point, a or b is at the higher potential? (b) Calculate the value of E. (c) A negative point charge \(q = -0.200 \mu C\) is moved from b to a. Calculate the work done on the point charge by the electric field. Text Transcription: q = -0.200 muC

Problem 21DQ In electronics it is customary to define the potential of ground (thinking of the earth as a large conductor) as zero. Is this consistent with the fact that the earth has a net electric charge that is not zero? (Refer to Exercise 21.28.) 21.28 Electric Field of the Earth. The earth has a net electric charge that causes a field at points near its surface equal to 150 N/C and directed in toward the center of the earth. (a) What magnitude and sign of charge would a 60-kg human have to acquire to overcome his or her weight by the force exerted by the earth’s electric field? (b) What would be the force of repulsion between two people each with the charge calculated in part (a) and separated by a distance of 100 m? Is use of the earth’s electric field a feasible means of flight? Why or why not?

A positive charge q is fixed at the point x = 0, y = 0, and a negative charge -2q is fixed at the point x = a, y = 0. (a) Show the positions of the charges in a diagram. (b) Derive an expression for the potential V at points on the x-axis as a function of the coordinate x. Take V to be zero at an infinite distance from the charges. (c) At which positions on the x-axis is V = 0? (d) Graph V at points on the x-axis as a function of x in the range from x = -2a to x = +2a. (e) What does the answer to part (b) become when x >> a? Explain why this result is obtained.

Problem 22E Consider the arrangement of point charges described in Exercise (a) Derive an expression for the potential v at points on the y-axis as a function of the coordinate y take v to be zero at an infinite distance from the charges. (b) At which position on the y-axis is V = 0? (c) Graph V at points on the y-axis as a function of y in the range from y = ?2a to y = +2a. What does the answer to part (a) become when y > a? Explain result is obtained. Exercise: A positive charge q is fixed at the point x = O, y = O, and a negative charge ?2q is fixed at the point x = a, y = O. a) Show the positions of the charges in a diagram. (b) Derive an expression for the potential V at points on the x-axis as a function of the coordinate x. Take V to be zero at an infinite distance from the charges. (c) At which positions on the x-axis is V = O? (d) Graph V at points on the x-axis as a function of x in the range from x = ?2a to x = +2a. (e) What does the answer to part (b) become when x ? a? Explain why this result is obtained.

Problem 33E A very long insulating cylindrical shell of radius 6.00 cm carries charge of linear density 8.50 µC/m spread uniformly over its outer surface. What would a voltmeter read if it were connected between (a) the surface of the cylinder and a point 4.00 cm above the surface, and (b) the surface and a point 1.00 cm from the central axis of the cylinder?

Problem 34E A ring of diameter 8.00 cm is fixed in place and carries a charge of +5.00 µC uniformly spread over its circumference. (a) How much work does it take to move a tiny + 3.00-µC charged ball of mass 1.50 g from very far away to the center of the ring? (b) Is it necessary to take a path along the axis of the ring? Why? (c) If the ball is slightly displaced from the center of the ring, what will it do and what is the maximum speed it will reach?

Problem 35E A very small sphere with positive charge q = +8.00 µC is released from rest at a point 1.50 cm from a very long line of uniform linear charge density ? = +3.00 µC/m. What is the kinetic energy of the sphere when it is 4.50 cm from the line of charge if the only force on it is the force exerted by the line of charge?

Problem 42E (a) How much excess charge must be placed on a copper sphere 25.0 cm in diameter so that the potential of its center, relative to infinity, is 1.50 kV? (b) What is the potential of the sphere’s surface relative to infinity?

The electric field at the surface of a charged, solid, copper sphere with radius 0.200 m is 3800 N/C, directed toward the center of the sphere. What is the potential at the center of the sphere, if we take the potential to be zero infinitely far from the sphere?

Problem 44E A very large plastic sheet carries a uniform charge density of -6.00 nC/m2 on one face. (a) As you move away from the sheet along a line perpendicular to it, does the potential increase or decrease? How do you know, without doing any calculations? Does your answer depend on where you choose the reference point for potential? (b) Find the spacing between equipotential surfaces that differ from each other by 1.00 V. What type of surfaces are these?

Problem 45E CALC In a certain region of space, the electric potential is V(x, y, z) = Axy - Bx2 + Cy, where A, B, and C are positive constants. (a) Calculate the x-, y-, and z-components of the electric field. (b) At which points is the electric field equal to zero?

Problem 46E CALC In a certain region of space the electric potential is given by V = +Ax2y – Bxy2, where A = 5.00 V/m3 and B = 8.00 V/m3. Calculate the magnitude and direction of the electric field at the point in the region that has coordinates x = 2.00 m, y = 0.400 m, and z = 0.

A metal sphere with radius is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius . There is charge on the inner sphere and charge on the outer spherical shell. (a) Calculate the potential \(v(r) f \circ r(i) y<r_{a} ; \text { (ii) }_{a}<r<r_{b} ;(i i) r>r_{b}\). (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take to be zero when is infinite. (b) Show that the potential of the inner sphere with respect to the outer is \(V_{a b}=\frac{9}{4 \pi \epsilon_{0}}\left(\frac{1}{r_{a}}-\frac{1}{r_{b}}\right)\) (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude \(E(r)=\frac{V_{a b}}{\left(1 / Y_{a}-1 / r_{b}\right)} \frac{1}{r^{2}}\) (d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distance from the center, where \(\gamma>\gamma_{b}\) (e) Suppose the charge on the outer sphere is not but a negative charge of different magnitude, say . Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different. Equation transcription: Text transcription: v(r) f \circ r(i) y<r{a} ; \text { (ii) }{a}<r<r{b} ;(i i) r>r{b} V{a b}=\frac{9}{4 \pi epsilon_{0}}\left(\frac{1}{r{a}}-\frac{1}{r{b}}) E(r)=frac{V{a b}}{\left(1 / Y{a}-1 / r{b})} \frac{1}{r^{2}} \gamma>\gamma_{b}

A metal sphere with radius \(r_{a}=1.20 \mathrm{~cm}\) is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius \(r_{b}=9.60 \mathrm{~cm}\) Charge is put on the inner sphere and charge on the outer spherical shell. The magnitude of is chosen to make the potential difference between the spheres , with the inner sphere at higher potential. (a) Use the result of Exercise to calculate (b) With the help of the result of Exercise , sketch the equipotential surfaces that correspond to , and . (c) In your sketch, show the electric field lines. Are the electric field lines and equipotential surfaces mutually perpendicular? Are the equipotential surfaces closer together when the magnitude of \(\bar{E}\) is largest? Equation transcription: Text transcription: r_{a}=1.20 \mathrm{~cm} r_{b}=9.60 \mathrm{~cm} \bar{E}

Problem 49E A very long cylinder of radius 2.00 cm carries a uniform charge density of 1.50 nC/m. (a) Describe the shape of the equipotential surfaces for this cylinder. (b) Taking the reference level for the zero of potential to be the surface of the cylinder, find the radius of equipotential surfaces having potentials of 10.0 V, 20.0 V, and 30.0 V. (c) Are the equipotential surfaces equally spaced? If not, do they get closer together or farther apart as r increases?

Problem 50P CP A point charge q1 = +5.00 µC is held fixed in space. From a horizontal distance of 6.00 cm, a small sphere with mass 4.00 X 10-3 kg and charge q2 = +2.00 µC is fired toward the fixed charge with an initial speed of 40.0 m/s. Gravity can be neglected. What is the acceleration of the sphere at the instant when its speed is 25.0 m/s?

Problem 51P A point charge q1 = 4.00 nC is placed at the origin, and a second point charge q2 = -3.00 nC is placed on the x-axis at x = +20.0 cm. A third point charge q3 = 2.00 nC is to be placed on the x-axis between q1 and q2. (Take as zero the potential energy of the three charges when they are infinitely far apart.) (a) What is the potential energy of the system of the three charges if q3 is placed at x = + 10.0 cm? (b) Where should q3 be placed to make the potential energy of the system equal to zero?

A small sphere with mass 5.00 \(\times\) 107 kg and change + 3.00?C is released from rest a distance of 0.400 m above a horizontal insulating sheet of charge that has uniform surface charge density ? = +8.00 pC/m2. Using energy calculate the speed of the sphere when it is 0.100 m above the sheet of charge?

Determining the Size of the Nucleus. When radium- 226 decays radioactively, it emits an alpha particle (the nucleus of helium), and the end product is radon-222. We can model this decay by thinking of the radium-226 as consisting of an alpha particle emitted from the surface of the spherically symmetric radon- 222 nucleus, and we can treat the alpha particle as a point charge. The energy of the alpha particle has been measured in the laboratory and has been found to be 4.79 MeV when the alpha particle is essentially infinitely far from the nucleus. Since radon is much heavier than the alpha particle, we can assume that there is no appreciable recoil of the radon after the decay. The radon nucleus contains 86 protons, while the alpha particle has 2 protons and the radium nucleus has 88 protons. (a) What was the electric potential energy of the alpha–radon combination just before the decay, in MeV and in joules? (b) Use your result from part (a) to calculate the radius of the radon nucleus.

A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it (Fig. P23.64). A large potential difference is established between the wire and the outer cylinder, with the wire at higher potential; this sets up a strong electric field directed radially outward. When ionizing radiation enters the device, it ionizes a few air molecules. The free electrons produced are accelerated by the electric field toward the wire and, on the way there, ionize many more air molecules. Thus a current pulse is produced that can be detected by appropriate electronic circuitry and converted to an audible "click." Suppose the radius of the central wire is \(145 \mu C\) and the radius of the hollow cylinder is What potential difference between the wire and the cylinder produces an electric field of \(2.00 \times 10^{4} \mathrm{~V} / \mathrm{m}\) at a distance of from the axis of the wire? (The wire and cylinder are both very long in comparison to their radii, so the results of Problem apply.) Equation transcription: Text transcription: 145 \mu C 2.00 \times 10^{4}{~V} /{m}

CP Deflection in a CRT. Cathode-ray tubes (CRTs) are often found in oscilloscopes and computer monitors. In Fig. P23.65 an electron with an initial speed of \(6.50 \times 10^{6} \mathrm{~m} / \mathrm{s}\) is projected along the axis midway between the deflection plates of a cathode ray tube. The potential difference between the two plates is and the lower plate is the one at higher potential. (a) What is the force (magnitude and direction) on the electron when it is between the plates? (b) What is the acceleration of the electron (magnitude and direction) when acted on by the force in part (a)? (c) How far below the axis has the electron moved when it reaches the end of the plates? (d) At what angle with the axis is it moving as it leaves the plates? (e) How far below the axis will it strike the fluorescent screen S? Equation transcription: Text transcription: 6.50 \times 10^{6}{~m} /{s}

Problem 66P CP Deflecting Plates of an Oscilloscope. The vertical deflecting plates of a typical classroom oscilloscope are a pair of parallel square metal plates carrying equal but opposite charges. Typical dimensions are about 3.0 cm on a side, with a separation of about 5.0 mm. The potential difference between the plates is 25.0 V. The plates are close enough that we can ignore fringing at the ends. Under these conditions: (a) how much charge is on each plate, and (b) how strong is the electric field between the plates? (c) If an electron is ejected at rest from the negative plate, how fast is it moving when it reaches the positive plate?

Problem 73P Charge Q = +4.00 µC is distributed uniformly over the volume of an insulating sphere that has radius R = 5.00 cm. What is the potential difference between the center of the sphere and the surface of the sphere?

Problem 74P An insulating spherical shell with inner radius 25.0 cm and outer radius 60.0 cm carries a charge of +150.0 µC uniformly distributed over its outer surface. Point a is at the center of the shell, point b is on the inner surface, and point c is on the outer surface. (a) What will a voltmeter read if it is connected between the following points: (i) a and b; (ii) b and c ; (iii) c and infinity; (iv) a and c? (b) Which is at higher potential: (i) a or b; (ii) b or c; (iii) a or c? (c) Which, if any, of the answers would change sign if the charge were -150 µC?

Problem 75P Exercise shows that, outside a spherical shell with uniform surface charge, the potential is the same as if all the charge were concentrated into a point charge at the center of the sphere. (a) Use this result to show that for two uniformly charged insulating shells, the force they exert on each other and their mutual electrical energy are the same as if all the charge were concentrated at their centers. (Hint: See Section 13.6.) (b) Does this same result hold for solid insulating spheres, with charge distributed uniformly throughout their volume? (c) Does this same result hold for the force between two charged conducting shells? Between two charged solid conductors? Explain. Exercise: (a) Show that V for a spherical shell of radius R, that has charge q distributed uniformly over its surface, is the same as V for a solid conductor with radius R and charge q. (b) You rub an inflated balloon on the carpet and it acquires a potential that is 1560 V lower than its potential before it became charged. If the charge is uniformly distributed over the surface of the balloon and if the radius of the balloon is 15 cm, what is the net charge on the balloon? (e) In light of its 1200-V potential difference relative to you, do you think this balloon is dangerous? Explain.

Problem 76P CP Two plastic spheres, each carrying charge uniformly distributed throughout its interior, are initially placed in contact and then released. One sphere is 60.0 cm in diameter, has mass 50.0 g, and contains -10.0 µC of charge. The other sphere is 40.0 cm in diameter, has mass 150.0 g, and contains -30.0 µC of charge. Find the maximum acceleration and the maximum speed achieved by each sphere (relative to the fixed point of their initial location in space), assuming that no other forces are acting on them. (Hint: The uniformly distributed charges behave as though they were concentrated at the centers of the two spheres.)

Problem 77P Use the electric field calculated to calculate the potential difference between the solid conducting sphere and the thin insulating shell. Problem: A solid conducting sphere with radius R that carries positive charge Q is concentric with a very thin insulating shell of radius 2R that also carries charge Q. The charge Q is distributed uniformly over the insulating shell. (a) Find the electric held (magnitude and direction) in each of the regions 0 < r < R, R < r < 2R, and r > 2R. (b) Graph the electric-field magnitude as a function of r.

Consider a solid conducting sphere inside a hollow conducting sphere, with radii and charges specified in Problem 22.44. Take V = 0 as \(r \rightarrow \infty\). Use the electric field calculated in Problem 22.44 to calculate the potential V at the following values of r: (a) r = c (at the outer surface of the hollow sphere); (b) r = b (at the inner surface of the hollow sphere); (c) r = a (at the surface of the solid sphere); (d) r = 0 (at the center of the solid sphere).

CALC Electric charge is distributed uniformly along a thin rod of length a, with total charge Take the potential to be zero at infinity. Find the potential at the following points (Fig. P23.79): (a) point a distance to the right of the rod, and (b) point a distance above the right-hand end of the rod. (c) In parts (a) and (b), what does your result reduce to as or becomes much larger than ?

(a) If a spherical raindrop of radius 0.650 mm carries a charge of -3.60 pC uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop.) (b) Two identical raindrops, each with radius and charge specified in part (a), collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

Problem 81P Two metal spheres of different sizes are charged such that the electric potential is the same at the surface of each. Sphere A has a radius three times that of sphere B. Let QA and QB be the charges on the two spheres, and let EA and EB be the electric-field magnitudes at the surfaces of the two spheres. What are (a) the ratio QB/QA and (b) the ratio EB/EA?

Problem 82P An alpha particle with kinetic energy 11.0 MeV makes a head-on collision with a lead nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the lead nucleus remains stationary and that it may be treated as a point charge. The atomic number of lead is 82. The alpha particle is a helium nucleus, with atomic number 2.)

A metal sphere with radius \(R_1\) has a charge \(Q_1\). Take the electric potential to be zero at an infinite distance from the sphere. (a) What are the electric field and electric potential at the surface of the sphere? This sphere is now connected by a long, thin conducting wire to another sphere of radius \(R_2\) that is several meters from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere; (c) the electric potential at the surface of each sphere; (d) the electric field at the surface of each sphere? Assume that the amount of charge on the wire is much less than the charge on each sphere.

CALC Use the charge distribution and electric field calculated in Problem 22.65. (a) Show that for \(r \geq R\) the potential is identical to that produced by a point charge (Take the potential to be zero at infinity.) (b) Obtain an expression for the electric potential valid in the region \(r \leq R\). Equation transcription: Text transcription: r geq R r leq R

CALC A hollow, thin-walled insulating cylinder of radius and length (like the cardboard tube in a roll of toilet paper) has charge uniformly distributed over its surface. (a) Calculate the electric potential at all points along the axis of the tube. Take the origin to be at the center of the tube, and take the potential to be zero at infinity. (b) Show that if , the result of part (a) reduces to the potential on the axis of a ring of charge of radius . (See Example in Section ) (c) Use the result of part (a) to find the electric field at all points along the axis of the tube.