Two point charges are fixed on the axis. (a) Each has a

Aproton is moved to the left in a uniform electric field that points to the right. Is the proton moving in the direction of increasing or decreasing electric potential? Is the electrostatic potential energy of the proton increasing or decreasing?

An electron is moved to the left in a uniform electric field that points to the right. Is the electron moving in the direction of increasing or decreasing electric potential? Is the electrostatic potential energy of the electron increasing or decreasing? SSM

If the electric potential is uniform throughout a region of space, what can be said about the electric field in that region?

If is known at only a single point in space, can be found at that point? Explain your answer.

Figure 23-29 shows a point particle that has a positive charge and a metal sphere that has a charge Sketch the electric field lines and equipotential surfaces for this system of charges. SSM

Figure 23-30 shows a point particle that has a negative charge and a metal sphere that has a charge Sketch the electric field lines and equipotential surfaces for this system of charges.

Sketch the electric field lines and equipotential surfaces for the region surrounding the charged conductor shown in Figure 23-31, assuming that the conductor has a net positive charge.

Two equal positive point charges are separated by a finite distance. Sketch the electric field lines and the equipotential surfaces for this system.

Two point charges are fixed on the axis. (a) Each has a positive charge One is at and the other is at At the origin, which of the following is true? (1) and (2) and (3) and (4) and (5) None of the above (b) One point charge has a positive charge and the other has a negative charge The positive point charge is at and the negative point charge is at At the origin, which of the following is true? (1) and (2) and (3) and (4) and (5) None of the above

The electrostatic potential (in volts) is given by where is a constant, and is in meters. (a) Sketch the electric field for this potential. (b) Which of the following charge distributions is most likely responsible for this potential: (1) A negatively charged flat sheet in the plane, (2) a point charge at the origin, (3) a positively charged flat sheet in the plane, or (4) a uniformly charged sphere centered at the origin? Explain your answer.

The electric potential is the same everywhere on the surface of a conductor. Does this mean that the surface charge density is also the same everywhere on the surface? Explain your answer.

Three identical positive point charges are located at the vertices of an equilateral triangle. If the length of each side of the triangle shrinks to one-fourth of its original length, by what factor does the electrostatic potential energy of this system change? (The electrostatic potential energy approaches zero if the length of each side of the triangle approaches infinity.)

Estimate the maximum potential difference between a thundercloud and Earth, given that the electrical breakdown of air occurs at fields of roughly

The specifications for the gap width of typical automotive spark plugs is approximately equal to the thickness of the cardboard used for matchbook covers. Because of the high compression of the airgas mixture in the cylinder, the dielectric strength of the mixture is roughly Estimate the maximum potential difference across the spark gap during operating conditions.

The radius of a proton is approximately Suppose two protons having equal and opposite momenta undergo a head-on collision. Estimate the minimum kinetic energy (in required by each proton to allow the protons to overcome electrostatic repulsion and collide. Hint: The rest energy of a proton is If the kinetic energies of the protons are much less than this rest energy, then a nonrelativistic calculation is justified.

When you touch a friend after walking across a rug on a dry day, you typically draw a spark of about Estimate the potential difference between you and your friend just before the spark.

Estimate the maximum surface charge density that can exist at the end of a sharp lightning rod so that no dielectric breakdown of air occurs.

The electric field strength near the surface of Earth is about (a) Estimate the magnitude of the charge density on the surface of Earth. (b) Estimate the total charge on Earth. (c) What is value of the electric potential at Earths surface? (Assume the potential is zero at infinity.) (d) If all Earths electrostatic potential energy could be harnessed and converted to electric energy at reasonable efficiency, how long could it be used to run the consumer households in the United States? Assume the average American household consumes about of electric energy per month.

Apoint particle has a charge equal to and is fixed at the origin. (a) What is the electric potential at a point from the origin assuming that at infinity? (b) How much work must be done to bring a second point particle that has a charge of from infinity to a distance of from the charge?

The facing surfaces of two large parallel conducting plates separated by have uniform surface charge densities that are equal in magnitude but opposite in sign. The difference in potential between the plates is (a) Is the positive plate or the negative plate at the higher potential? (b) What is the magnitude of the electric field between the plates? (c) An electron is released from rest next to the negatively charged surface. Find the work done by the electric field on the electron as the electron moves from the release point to the positive plate. Express your answer in both electron volts and joules. (d) What is the change in potential energy of the electron when it moves from the release point to the positive plate? (e) What is its kinetic energy when it reaches the positive plate?

A uniform electric field has a magnitude and points in the direction. (a) What is the electric potential difference between the plane and the plane? A point particle that has a charge of is released from rest at the origin. (b) What is the change in the electric potential energy of the particle as it travels from the plane to the plane? (c) What is the kinetic energy of the particle when it arrives at the plane? (d) Find the expression for the electric potential if its value is chosen to be zero at

In a potassium chloride unit, the distance between the potassium ion and the chloride ion is (a) Calculate the energy (in required to separate the two ions to an infinite distance apart. (Model the two ions as two point particles initially at rest.) (b) If twice the energy determined in Part (a) is actually supplied, what is the total amount of kinetic energy that the two ions have when they were an infinite distance apart?

Protons are released from rest in a Van de Graaff accelerator system. The protons initially are located where the electric potential has a value of and then they travel through a vacuum to a region where the potential is zero. (a) Find the final speed of these protons. (b) Find the accelerating electric field strength if the potential changed uniformly over a distance of

The picture tube of a television set was, until recently, invariably a cathode-ray tube. In a typical cathode-ray tube, an electron gun arrangement is used to accelerate electrons from rest to the screen. The electrons are accelerated through a potential difference of (a) Which region is at a higher electric potential, the screen or the electrons starting location? Explain your answer. (b) What is the kinetic energy (in both and of an electron as it reaches the screen?

(a) A positively charged particle is on a trajectory to collide head-on with a massive positively charged nucleus that is initially at rest. The particle initially has kinetic energy In addition, the particle is initially far from the nucleus. Derive an expression for the distance of closest approach. Your expression should be in terms of the initial kinetic energy of the particle, the charge on the particle, and the charge on the nucleus, where both and are integers. (b) Find the numerical value for the distance of closest approach between a particle and a stationary gold nucleus and between a particle and a stationary gold nucleus. (The values and are the initial kinetic energies of the alpha particles. Neglect the motion of the gold nucleus following the collisions.) (c) The radius of the gold nucleus is about If particles approach the nucleus closer than they experience the strong nuclear force in addition to the electric force of repulsion. In the early twentieth century, before the strong nuclear force was known, Ernest Rutherford bombarded gold nuclei with particles that had kinetic energies of about Would you expect this experiment to reveal the existence of this strong nuclear force? Explain your answer.

Four point charges, each having a magnitude of are fixed at the corners of a square whose edges are long. Find the electric potential at the center of the square if (a) all the charges are positive, (b) three of the charges are positive and one charge is negative, and (c) two charges are positive and two charges are negative. (Assume the potential is zero very far from all charges.)

Three point charges are fixed at locations on the axis: is at is at and is at Find the electric potential at the point on the axis at if (a) (b) and and (c) and (Assume the potential is zero very far from all charges.)

Points and are fixed at the vertices of an equilateral triangle whose edges are long. A point particle that has a charge of is fixed at each of vertices and (a) What is the electric potential at point (Assume the potential is zero very far from all charges.) (b) How much work is required to move a point particle having a charge of from a distance of infinity to point (c) How much additional work is required to move the point particle from point to the midpoint of side

Three identical point particles that have charge are at the vertices of an equilateral triangle that is circumscribed by a circle of radius that lies in the plane and is centered at the origin. The values of and are and respectively. (Assume the potential is zero very far from all charges.) (a) What is the electric potential at the origin? (b) What is the electric potential at the point on the axis at (c) How would your answers to Parts (a) and (b) change if the charges were still on the circle but one is no longer at a vertex of the triangle? Explain your answer.

Two point charges and are separated by a distance At a point from and along the line joining the two charges the potential is zero. (Assume the potential is zero very far from all charges.) (a) Which of the following statements is true? (1) The charges have the same sign. (2) The charges have opposite signs. (3) The relative signs of the charges cannot be determined by using the data given. (b) Which of the following statements is true? (1) (2) (3) (4) The relative magnitudes of the charges cannot be determined by using the data given. (c) Find the ratio

Two identical positively charged point particles are fixed on the axis at and (a)Write an expression for the electric potential as a function of for all points on the axis. (b) Sketch versus for all points on the axis.

A point charge of is at the origin and a second point charge of is on the axis at (a) Sketch the potential function versus for all points on the axis. (b) At what point or points, if any, is on the axis? (c) At what point or points, if any, on the axis is the electric field zero? Are these locations the same locations found in Part (b) ? Explain your answer. (d) How much work is needed to bring a third charge to the point on the axis?

A dipole consists of equal but opposite point charges and It is located so that its center is at the origin, and its axis is aligned with the axis (Figure 23-32). The distance between the charges is Let be the vector from the origin to an arbitrary field point and be the angle that makes with the direction. (a) Show that at large distances from the dipole (i.e., for the dipoles electric potential is given by where is the dipole moment of the dipole and is the angle between and (b) At what points in the region other than at infinity, is the electric potential zero? SSM

A charge configuration consists of three point charges located on the z axis (Figure 23-33). One has a charge equal to -2q, and is located at the origin. The other two each have a charge equal to +q, one is located at z=+L and the other is located at z=-L. This charge configuration can be modeled as two dipoles: one centered at z=+L/2 and with a dipole moment in the +z direction, the other centered at z=-L/2 and with a dipole moment in the -z direction. Each of these dipoles has a dipole moment that has a magnitude equal to qL. Two dipoles arranged in this fashion form a linear electric quadrupole. (There are other geometrical arrangements of dipoles that create quadrupoles but they are not linear.) (a) Using the result from Problem 33, show that at large distances from the quadrupole (i.e., for \(r \gg L\) ), the electric potential is given by \(V_{\text {quad }}(r, \theta)=2 k B \cos ^2 \theta / r^3\), where \(B=q L^2\). ( B is the magnitude of the quadrupole moment of the charge configuration.) (b) Show that on the positive z axis, this potential gives an electric field (for \(z \gg L\) ) of \(\vec{E}=\left(6 k B / z^4\right) \hat{k}\). (c) Show you get the result of Part (b) by adding the electric fields from the three point charges.

A uniform electric field is in the direction. Points and are on the axis, with at and at (a) Is the potential difference positive or negative? (b) If is what is the magnitude of the electric field?

An electric field is given by the expression where Find the potential difference between the point at and the point . Which of these points is at the higher potential?

The electric field on the axis due to a point charge fixed at the origin is given by where and (a) Find the magnitude and sign of the point charge. (b) Find the potential difference between the points on the axis at and Which of these points is at the higher potential?

The electric potential due to a particular charge distribution is measured at many points along the axis. A plot of the data is shown in Figure 23-34. At what location (or locations) is the component of the electric field equal to zero? At this location (or these locations) is the potential also equal to zero? Explain your answer.

Three identical point charges, each with a charge equal to lie in the plane. Two of the charges are on the axis at and and the third charge is on the axis at (a) Find the potential as a function of position along the axis. (b) Use the Part (a) result to obtain an expression for the component of the electric field as a function of Check your answers to Parts (a) and (b) at the origin and as approaches to see if they yield the expected results.

A charge of is uniformly distributed on a thin spherical shell of radius (Assume the potential is zero very far from all charges.) (a) What is the magnitude of the electric field just outside and just inside the shell? (b) What is the magnitude of the electric potential just outside and just inside the shell? (c) What is the electric potential at the center of the shell? (d) What is the magnitude of the electric field at the center of the shell?

An infinite line charge of linear charge density lies on the axis. Find the electric potential at distances from the line charge of (a) (b) and (c) Assume that we choose at a distance of from the line of charge.

(a) Find the maximum net charge that can be placed on a spherical conductor of radius before dielectric breakdown of the air occurs. (b) What is the electric potential of the sphere when it has this maximum charge? (Assume the potential is zero very far from all charges.)

Find the maximum surface charge density that can exist on the surface of any conductor before dielectric breakdown of the air occurs.

A conducting spherical shell of inner radius and outer radius is concentric with a small metal sphere of radius The metal sphere has a positive charge The total charge on the conducting spherical shell is (Assume the potential is zero very far from all charges.) (a) What is the electric potential of the spherical shell? (b) What is the electric potential of the metal sphere?

Two coaxial conducting cylindrical shells have equal and opposite charges. The inner shell has charge and an outer radius and the outer shell has charge and an inner radius The length of each cylindrical shell is and is very long compared with Find the potential difference between the shells.

Positive charge is placed on two conducting spheres that are very far apart and connected by a long, very thin conducting wire. The radius of the smaller sphere is and the radius of the larger sphere is The electric field strength at the surface of the larger sphere is Estimate the surface charge density on each sphere.

Two concentric conducting spherical shells have equal and opposite charges. The inner shell has outer radius and charge the outer shell has inner radius and charge Find the potential difference between the shells.

The electric potential at the surface of a uniformly charged sphere is At a point outside the sphere at a (radial) distance of from its surface, the electric potential is (The potential is zero very far from the sphere.) What is the radius of the sphere, and what is the charge of the sphere?

Consider two infinite parallel thin sheets of charge, one in the plane and the other in the plane. The potential is zero at the origin. (a) Find the electric potential everywhere in space if the planes have equal positive charge densities (b) Find the electric potential everywhere in space if the sheet in the plane has a charge density and the sheet in the plane has a charge density

The expression for the potential along the axis of a thin uniformly charged disk is given by (Equation 23-20), where and are the radius and the charge per unit area of the disk, respectively. Show that this expression reduces to for where is the total charge on the disk. Explain why this result is expected. Hint: Use the binomial theorem to expand the radical.

A rod of length has a total charge uniformly distributed along its length. The rod lies along the axis with its center at the origin. (a) Find an expression for the electric potential as a function of position along the axis. (b) Show that the result obtained in Part (a) reduces to for Explain why this result is expected.

A rod of length has a charge uniformly distributed along its length. The rod lies along the axis with one end at the origin. (a) Find an expression for the electric potential as a function of position along the axis. (b) Show that the result obtained in Part (a) reduces to for Explain why this result is expected.

Adisk of radius has a surface charge distribution given by where is a constant and is the distance from the center of the disk. (a) Find the total charge on the disk. (b) Find an expression for the electric potential at a distance from the center of the disk on the axis that passes through the disks center and is perpendicular to its plane.

Adisk of radius has a surface charge distribution given by where is a constant and is the distance from the center of the disk. (a) Find the total charge on the disk. (b) Find an expression for the electric potential at a distance from the center of the disk on the axis that passes through the disks center and is perpendicular to its plane.

A rod of length has a total charge uniformly distributed along its length. The rod lies along the axis with its center at the origin. (a) What is the electric potential as a function of position along the axis for (b) Show that for your result reduces to that due to a point charge

A circle of radius is removed from the center of a uniformly charged thin circular disk of radius and charge per unit area (a) Find an expression for the potential on the axis a distance from the center of the disk. (b) Show that for the electric potential on the axis of the uniformly charged disk with cutout approaches where is the total charge on the disk.

The expression for the electric potential inside a uniformly charged solid sphere is given by where is the radius of the sphere and is the distance from the center. This expression was obtained in Example 23-12 by first finding the electric field. In this problem, you derive the same expression by modeling the sphere as a nested collection of thin spherical shells, and then adding the potentials of these shells at a field point inside the sphere. The potential that is a distance from the center of a uniformly charged thin spherical shell that has a radius and a charge is given by for and for (Equation 23-22). Consider a sphere of radius containing a charge that is uniformly distributed and you want to find at some point inside the sphere (i.e., for (a) Find an expression for the charge on a spherical shell of radius and thickness (b) Find an expression for the potential at due to the charge on a shell of radius and thickness where (c) Integrate your expression in Part (b) from to to find the potential at due to all the charge in the region farther than from the center of the sphere. (d) Find an expression for the potential at due to the charge in a shell of radius and thickness where (e) Integrate your expression in Part (d) from to to find the potential at due to all the charge in the region closer than to the center of the sphere. (f) Find the total potential at by adding your Part (c) and Part (e) results.

Calculate the electric potential at the point a distance from the center of a uniformly charged thin spherical shell of radius and charge (Assume the potential is zero far from the shell.)

A circle of radius is removed from the center of a uniformly charged thin circular disk of radius Show that the potential at a point on the central axis of the disk a distance from its geometrical center is given by where is the charge density of the disk.

An infinite flat sheet of charge has a uniform surface charge density equal to How far apart are the equipotential surfaces whose potentials differ by

Consider two parallel uniformly charged infinite planes that are equal but oppositely charged. (a) What is (are) the shape(s) of the equipotential surfaces in the region between them? Explain your answer. (b) What is (are) the shape(s) of the equipotential surfaces in the regions not between them? Explain your answer. SSM 100 V?

AGeiger tube consists of two elements, a long metal cylindrical shell and a long straight metal wire running down its central axis. Model the tube as if both the wire and cylinder are infinitely long. The central wire is positively charged and the outer cylinder is negatively charged. The potential difference between the wire and the cylinder is (a) What is the direction of the electric field inside the tube? (b) Which element is at a higher electric potential? (c) What is (are) the shape(s) of the equipotential surfaces inside the tube? (d) Consider two equipotential surfaces described in Part (c). Suppose they differ in electric potential by Do two such equipotential surfaces near the central wire have the same spacing as they would near the outer cylinder? If not, where in the tube are the equipotential surfaces that are more widely spaced? Explain your answer.

Suppose the cylinder in the Geiger tube in Problem 62 has an inside diameter of and the wire has a diameter of The cylinder is grounded so its potential is equal to zero. (a) What is the radius of the equipotential surface that has a potential equal to Is this surface closer to the wire or to the cylinder? (b) How far apart are the equipotential surfaces that have potentials of (c) Compare your result in Part (b) to the distance between the two surfaces that have potentials of respectively. What does this comparison tell you about the electric field strength as a function of the distance from the central wire?

Apoint particle that has a charge of is at the origin. (a) What is (are) the shapes of the equipotential surfaces in the region around this charge? (b) Assuming the potential to be zero at calculate the radii of the five surfaces that have potentials equal to and sketch them to scale centered on the charge. (c) Are these surfaces equally spaced? Explain your answer. (d) Estimate the electric field strength between the and equipotential surfaces by dividing the difference between the two potentials by the difference between the two radii. Compare this estimate to the exact value at the location midway between these two surfaces.

Three point charges are on the axis: is at the origin, is at and is at Find the electrostatic potential energy of this system of charges for the following charge values: (a) (b) and and (c) and (Assume the potential energy is zero when the charges are very far from each other.)

Point charges and are fixed at the vertices of an equilateral triangle whose sides are long. Find the electrostatic potential energy of this system of charges for the following charge values: (a) (b) and and (c) and (Assume the potential energy is zero when the charges are very far from each other.)

(a) How much charge is on the surface of an isolated spherical conductor that has a radius and is charged to (b) What is the electrostatic potential energy of this conductor? (Assume the potential is zero far from the sphere.)

Four point charges, each having a charge with a magnitude of are at the corners of a square whose sides are long. Find the electrostatic potential energy of this system under the following conditions: (a) all of the charges are negative, (b) three of the charges are positive and one of the charges is negative, (c) the charges at two adjacent corners are positive and the other two charges are negative, and (d) the charges at two opposite corners are positive and the other two charges are negative. (Assume the potential energy is zero when the point charges are very far from each other.)

Four point charges are fixed at the corners of a square centered at the origin. The length of each side of the square is The charges are located as follows: is at is at is at and is at A fifth particle that has a mass and a charge is placed at the origin and released from rest. Find its speed when it is a very far from the origin.

Consider two point particles that have charge are at rest, and are separated by (a) How much work was required to bring them together from a very large separation distance? (b) If they are released, how much kinetic energy will each have when they are separated by twice their separation at release? (c) The mass of each particle is What speed will each have when they are very far from each other?

Consider an electron and a proton that are initially at rest and are separated by Neglecting any motion of the much more massive proton, what is the minimum (a) kinetic energy and (b) speed with which the electron must be projected at so it reaches a point a distance of from the proton? Assume the electrons velocity is directed radially away from the proton. (c) How far will the electron travel away from the proton if it has twice that initial kinetic energy?

A positive point charge equal to \(4.80 \times 10^{-19} \ \mathrm C\) is separated from a negative point charge of the same magnitude by \(6.40 \times 10^{-10} \ \mathrm m\). What is the electric potential at a point \(9.20 \times 10^{-10} \ \mathrm m\) from each of the two charges?

Two positive point charges each have a charge of and are fixed on the axis at and (a) Find the electric potential at any point on the axis. (b) Use your result in Part (a) to find the electric field at any point on the axis.

If a conducting sphere is to be charged to a potential of what is the smallest possible radius of the sphere so that the electric field near the surface of the sphere will not exceed the dielectric strength of air?

SPREADSHEET Two infinitely long parallel wires have a uniform charge per unit length and respectively. The wires are parallel with the axis. The positively charged wire intersects the axis at and the negatively charged wire intersects the axis at (a) Choose the origin as the reference point where the potential is zero, and express the potential at an arbitrary point in the plane in terms of and Use this expression to solve for the potential everywhere on the axis. (b) Using and obtain the equation for the equipotential surface in the plane that passes through the point (c) Use a spreadsheet program to plot the equipotential surface found in Part (b).

The equipotential curve graphed in Problem 75 should be a circle. (a) Show mathematically that it is a circle. (b) The equipotential circle in the plane is the intersection of a threedimensional equipotential surface and the plane. Describe the three-dimensional surface using one or two sentences.

The hydrogen atom in its ground state can be modeled as a positive point charge of magnitude (the proton) surrounded by a negative charge distribution that has a charge density (the electron) that varies with the distance from the center of the proton as (a result obtained from quantum mechanics), where is the most probable distance of the electron from the proton. (a) Calculate the value of needed for the hydrogen atom to be neutral. (b) Calculate the electrostatic potential (relative to infinity) of this system as a function of the distance r from the proton.

Charge is supplied to the metal dome of a Van de Graaff generator by the belt at the rate of when the potential difference between the belt and the dome is The dome transfers charge to the atmosphere at the same rate, so the potential difference is maintained. What minimum power is needed to drive the moving belt and maintain the potential difference?

A positive point charge is located on the axis at (a) How much work is required to bring an identical point charge from infinity to the point on the axis at (b) With the two identical point charges in place at and how much work is required to bring a third point charge from infinity to the origin? (c) How much work is required to move the charge from the origin to the point on the axis at along the semicircular path shown (Figure 23-35)?

A charge of is uniformly distributed on a ring of radius that lies in the plane and is centered at the origin. A point charge of is initially located on the axis at Find the work required to move the point charge to the origin.

Two metal spheres each have a radius of The centers of the two spheres are apart. The spheres are initially neutral, but a charge is transferred from one sphere to the other, creating a potential difference between the spheres of A proton is released from rest at the surface of the positively charged sphere and travels to the negatively charged sphere. (a) What is the protons kinetic energy just as it strikes the negatively charged sphere? (b) At what speed does it strike the sphere?

SPREADSHEET (a) Using a spreadsheet program, graph versus for a uniformly charged ring in the plane and centered at the origin. The potential on the axis is given by (Equation 23-19). (b) Use your graph to estimate the points on the axis where the electric field strength is greatest.

A spherical conductor of radius is charged to When it is connected by a long, very thin conducting wire to a second conducting sphere far away, its potential drops to What is the radius of the second sphere?

Ametal sphere centered at the origin has a surface charge density that has a magnitude of and a radius less than Adistance of from the origin, the electric potential is and the electric field strength is (Assume the potential is zero very far from the sphere.) (a) What is the radius of the metal sphere? (b) What is the sign of the charge on the sphere? Explain your answer.

Along the central axis of a uniformly charged disk, at a point from the center of the disk, the potential is and the magnitude of the electric field is At a distance of the potential is and the magnitude of the electric field is (Assume the potential is zero very far from the sphere.) Find the total charge on the disk.

A radioactive nucleus emits an particle that has a charge When the particle is a large distance from the nucleus, it has a kinetic energy of Assume that the particle had negligible kinetic energy as it left the surface of the nucleus. The daughter (or residual) nucleus has a charge Determine the radius of the nucleus. (Neglect the radius of the particle and assume the nucleus remains at rest.)

(a) Configuration consists of two point particles; one particle has a charge of and is on the axis at and the other particle has a charge of and is at (Figure 23-36a). Assuming the potential is zero at large distances from these charged particles, show that the potential is also zero everywhere on the plane. (b) Configuration consists of a flat metal plate of infinite extent and a point particle located a distance from the plate (Figure 23-36b). The point particle has a charge equal to and the plate is grounded. (Grounding the plate forces its potential to equal zero.) Choose the line perpendicular to the plate and through the point charge as the axis, and choose the origin at the surface of the plate nearest the particle. (These choices put the particle on the axis at For configuration the electric potential is zero both at all points in the half-space that are very far from the particle and at all points on the planejust as was the case for configuration A theorem, called the uniqueness theorem, implies that throughout the half-space the potential function and thus the electric field for the two configurations are identical. Using this result, obtain the electric field at every point in the plane in configuration (The uniqueness theorem tells us that in configuration the electric field at each point in the plane is the same as it is in configuration ) Use this result to find the surface charge density at each point in the conducting plane (in configuration B). SSM

Aparticle that has a mass and a positive charge is constrained to move along the axis. At and are two ring charges of radius (Figure 23-37). Each ring is centered on the axis and lies in a plane perpendicular to it. Each ring has a total positive charge uniformly distributed on it. (a) Obtain an expression for the potential on the axis due to the charge on the rings. (b) Show that has a minimum at (c) Show that for the potential approaches the form (d) Use the result of Part (c) to derive an expression for the angular frequency of oscillation of the mass if it is displaced slightly from the origin and released. (Assume the potential equals zero at points far from the rings.)

Three concentric conducting thin spherical shells have radii so that Initially, the inner shell is uncharged, the middle shell has a positive charge and the outer shell has a charge (Assume the potential equals zero at points far from the shells.) (a) Find the electric potential of each of the three shells. (b) If the inner and outer shells are now connected by a conducting wire that is insulated as it passes through a small hole in the middle shell, what is the electric potential of each of the three shells, and what is the final charge on each shell?

Consider two concentric spherical thin metal shells of radii and where The outer shell has a charge but the inner shell is grounded. This means that the potential on the inner shell is the same as the potential at points far from the shells. Find the charge on the inner shell.

Show that the total work needed to assemble a uniformly charged sphere that has a total charge of and radius is given by Energy conservation tells us that this result is the same as the resulting electrostatic potential energy of the sphere. Hint: Let be the charge density of the sphere that has charge and radius Calculate the work to bring in charge from infinity to the surface of a uniformly charged sphere of radius and charge density (No additional work is required to smear throughout a spherical shell of radius thickness and charge density Why?)

(a) Use the result of Problem 91 to calculate the classical electron radius, the radius of a uniform sphere that has a charge and an electrostatic potential energy equal to the rest energy of the electron Comment on the shortcomings of this model for the electron. (b) Repeat the calculation in Part (a) for a proton using its rest energy of Experiments indicate the proton has an approximate radius of about Is your result close to this value?

(a) Consider a uniformly charged sphere that has radius and charge and is composed of an incompressible fluid, such as water. If the sphere fissions (splits) into two halves of equal volume and equal charge, and if these halves stabilize into uniformly charged spheres, what is the radius of each? (b) Using the expression for potential energy shown in Problem 91, calculate the change in the total electrostatic potential energy of the charged fluid. Assume that the spheres are separated by a large distance.

Problem 93 can be modified to be used as a very simple model for nuclear fission. When a nucleus absorbs a neutron, it can fission into the fragments and 2 neutrons. The has 92 protons, while has 54 protons and has 38 protons. Estimate the energy released during this fission process (in assuming that the mass density of the nucleus is constant and has a value of 4 _ 1017 kg>m3.