Solved: The separation between the plates of a

An electron is released from rest in a region of space with a nonzero electric eld. As the electron moves, does it experience an increasing or decreasing electric potential? Explain.

Auniform electric eld of magnitude points in the positive xdirection. Find the change in electric potential energy of a charge as it moves from the origin to the points (a) (0, 6.0 m); (b) (6.0 m, 0); and (c) (6.0 m, 6.0 m).

Auniform electric eld of magnitude points in the positive x direction. Find the change in electric potential between the origin and the points (a) (0, 6.0 m); (b) (6.0 m, 0); and (c) (6.0 m, 6.0 m).

Electric Potential Across a Cell MembraneIn a typical living cell, the electric potential inside the cell is 0.070 V lower than the electric potential outside the cell. The thickness of the cell membrane is . What are the magnitude and direction of the electric eld within the cell membrane?

A computer monitor accelerates electrons and directs them to the screen in order to create an image. If the accelerating plates are 1.05 cm apart, and have a potential difference of 25,500 V, what is the magnitude of the uniform electric eld between them?

Find the change in electric potential energy for an electron that moves from one accelerating plate to the other in the computer monitor described in the previous problem.

Find the change in electric potential energy for an electron that moves from one accelerating plate to the other in the computer monitor described in the previous problem.

When an ion accelerates through a potential difference of 2140 V, its electric potential energy decreases by . What is the charge on the ion?

The Electric Potential of the EarthThe Earth has a vertical electric eld with a magnitude of approximately 100 V/m near its surface. What is the magnitude of the potential difference between a point on the ground and a point on the same level as the top of the Washington Monument (555 ft high)?

Auniform electric eld with a magnitude of 6350 N/C points in the positive x direction. Find the change in electric potential energy when a charge is moved 5.50 cm in (a)the positivexdirection, (b)the negative xdirection, and (c)the positive y direction.

Aspark plug in a car has electrodes separated by a gap of 0.025 in. To create a spark and ignite the air-fuel mixture in the engine, an electric eld of is required in the gap. (a) What potential difference must be applied to the spark plug to initiate a spark? (b)If the separation between electrodes is increased, does the required potential difference increase, decrease, or stay the same? Explain. (c) Find the potential difference for a separation of 0.050 in.

A uniform electric eld with a magnitude of 1200 N/C points in the negative x direction, as shown in Figure 2020.(a) What is the difference in electric potential, between points A and B? (b) What is the difference in electric potential, , between points B and C? (c) What is the difference in electric potential, , between points C and A? (d) From the information given in this problem, is it possible to determine the value of the electric potential at point A? If so, determine ; if not, explain why.

A Charged Battery A typical 12-V car battery can deliver of charge. If the energy supplied by the battery could be converted entirely to kinetic energy, what speed would it give to a 1400-kg car that is initially at rest?

The Sodium Pump Living cells actively pump positive sodium ions from inside the cell to outside the cell. This process is referred to as pumping because work must be done on the ions to move them from the negatively charged inner surface of the membrane to the positively charged outer surface. Given that the electric potential is 0.070 V higher outside the cell than inside the cell, and that the cell membrane is thick, (a) calculate the work that must be done (in joules) to move one sodium ion from inside the cell to outside. (b)If the thickness of the cell membrane is increased, does your answer to part (a) increase, decrease, or stay the same? Explain. (It is estimated that as much as 20% of the energy we consume in a resting state is used in operating this sodium pump.)

The electric potential of a system as a function of position along the x axis is given in Figure 2021. (a) In which of the regions, 1, 2, 3, or 4, do you expect to be greatest? In which region does have its greatest magnitude? Explain. (b)Calculate the value of in each of the regions, 1, 2, 3, and 4.

Points Aand B have electric potentials of 332 V and 149 V, respectively. When an electron released from rest at point Aarrives at point C, its kinetic energy is . When the electron is released from rest at point B, however, its kinetic energy when it reaches point C is . What are (a) the electric potential at point C and (b) the kinetic energy ?

Predict/Explain An electron is released from rest in a region of space with a nonzero electric eld. (a) As the electron moves, does the electric potential energy of the system increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. Because the electron has a negative charge its electric potential energy doesnt decrease, as one might expect, but increases instead. II. As the electron begins to move, its kinetic energy increases. The increase in kinetic energy is equal to the decrease in the electric potential energy of the system. III. The electron will move perpendicular to the electric eld, and hence its electric potential energy will remain the same.

Calculate the speed of (a) a proton and (b) an electron after each particle accelerates from rest through a potential difference of 275 V.

The electrons in a TV picture tube are accelerated from rest through a potential difference of 25 kV. What is the speed of the electrons after they have been accelerated by this potential difference?

Find the potential difference required to accelerate protons from rest to 10% of the speed of light. (At this point, relativistic effects start to become signicant.)

Aparticle with a mass of 3.8 g and a charge of is released from rest at point Ain Figure 2020. (a) In which direction will this charge move? (b) What speed will it have after moving through a distance of 5.0 cm? The electric eld has a magnitude of 1200 N/C. (c) Suppose the particle continues moving for another 5.0 cm. Will its increase in speed for the second 5.0 cm be greater than, less than, or equal to its increase in speed in the rst 5.0 cm? Explain.

Aproton has an initial speed of . (a) What potential difference is required to bring the proton to rest? (b)What potential difference is required to reduce the initial speed of the proton by a factor of 2? (c) What potential difference is required to reduce the initial kinetic energy of the proton by a factor of 2?

In Figure 2022, it is given that . (a) What value must have if the electric potential at point Ais to be zero? (b) With the value for found in part (a), is the electric potential at point B positive, negative, or zero? Explain

The charge in Figure 2022 has the value . (a) What value must have if the electric potential at point B is to be zero? (b) With the value for found in part (a), is the electric potential at point Apositive, negative, or zero? Explain.

It is given that the electric potential is zero at the center of the square in Figure 2022. (a) If , what is the value of the charge ? (b) Is the electric potential at point A positive, negative, or zero? Explain. (c) Is the electric potential at point B positive, negative, or zero? Explain.

The electric potential 1.1 m from a point charge q is . What is the value of q?

A point of charge is at the origin. What is the electric potential at (a)(3.0 m, 0); (b) and(c) ?

The Bohr Atom The hydrogen atom consists of one electron and one proton. In the Bohr model of the hydrogen atom, the electron orbits the proton in a circular orbit of radius . What is the electric potential due to the proton at the electrons orbit?

How far must the point charges and be separated for the electric potential energy of the system to be

Four different arrangements of point charges are shown in Figure 2023. In each case the charges are the same distance from the origin. Rank the four arrangements in order of increasing electric potential at the origin, taking the potential at innity to be zero. Indicate ties where appropriate.

Point charges and are placed on the x axis at (11 m, 0) and , respectively. (a) Sketch the electric potential on the x axis for this system. (b) Your sketch should show one point on the x axis between the two charges where the potential vanishes. Is this point closer to the charge or closer to the charge? Explain. (c) Find the point referred to in part (b).

(a) In the previous problem, nd the point to the left of the negative charge where the electric potential vanishes. (b) Is the electric eld at the point found in part (a) positive, negative, or zero? Explain.

A dipole is formed by point charges and placed on the x axis at (0.25 m, 0) and , respectively. (a) Sketch the electric potential on the x axis for this system. (b)At what positions on the x axis does the potential have the value

A charge of is held xed at the origin. A second charge of is released from rest at the position (1.25 m, 0.570 m). (a) If the mass of the second charge is 2.16 g, what is its speed when it moves innitely far from the origin? (b) At what distance from the origin does the second charge attain half the speed it will have at innity?

A charge of is held xed at the origin. (a) If a charge with a mass of 3.20 g is released from rest at the position (0.925 m, 1.17 m), what is its speed when it is halfway to the origin? (b) Suppose the charge is released from rest at the point and . When it is halfway to the origin, is its speed greater than, less than, or equal to the speed found in part (a)? Explain. (c) Find the speed of the charge for the situation described in part (b).

Acharge of is located at (3.055 m, 4.501 m), and a charge of is located at . (a)Find the electric potential at the origin. (b) There is one point on the line connecting these two charges where the potential is zero. Find this point.

Figure 2024 shows three charges at the corners of a rectangle.(a)How much work must be done to move the charge to innity? (b) Suppose, instead, that we move the charge to innity. Is the work required in this case greater than, less than, or the same as when we moved the charge to innity? Explain. (c) Calculate the work needed to move the charge to innity.

How much work must be done to move the three charges in Figure 2024 innitely far from one another?

(a) Find the electric potential at point P in Figure 2025. (b)Suppose the three charges shown in Figure 2025 are held in place. Afourth charge, with a charge of and a mass of 4.71 g, is released from rest at point P. What is the speed of the fourth charge when it has moved innitely far away from the other three charges?

Asquare of side a has a charge at each corner. What is the electric potential energy of this system of charges?

Asquare of side ahas charges and alternating from one corner to the next, as shown in Figure 2026. Find the electric potential energy for this system of charges.

Predict/Explain Apositive charge is moved from one location on an equipotential to another point on the same equipotential. (a) Is the work done on the charge positive, negative, or zero? (b)Choose the bestexplanationfrom among the following: I. The electric eld is perpendicular to an equipotential, therefore the work done in moving along an equipotential is zero. Because the charge is positive the work done on it is also positive. III. It takes negative work to keep the positive charge from accelerating as it moves along the equipotential.

Predict/Explain (a) Is the electric potential at point 1 in Figure 2019 greater than, less than, or equal to the electric potential at point 3? (b) Choose the best explanation from among the following: I. The electric eld lines point to the right, indicating that the electric potential is greater at point 3 than at point 1. II. The value of the electric potential is large where the electric eld lines are close together, and small where they are widely spaced. Therefore, the electric potential is the same at points 1 and 3. III. The electric potential decreases as we move in the direction of the electric eld, as shown in Figure 203. Therefore, the electric potential is greater at point 1 than at point 3.

Predict/Explain Imagine sketching a large number of equipotential surfaces in Figure 2019, with a constant difference in electric potential between adjacent surfaces. (a) Would the equipotentials at point 2 be more closely spaced, be less closely spaced, or have the same spacing as equipotentials at point 1? (b) Choose the bestexplanationfrom among the following: I. When electric eld lines are close together, the corresponding equipotentials are far apart. II. Equipotential surfaces, by denition, always have equal spacing between them. III. The electric eld is more intense at point 2 than at point 1, which means the equipotential surfaces are more closely spaced in that region.

Two point charges are on the x axis. Charge 1 is and is located at ; charge 2 is and is located at . Make sketches of the equipotential surfaces for this system (a) out to a distance of about 2.0 m from the origin and (b) far from the origin. In each case, indicate the direction in which the potential increases.

Two point charges are on the xaxis. Charge 1 is and is located at ; charge 2 is and is located at . Make sketches of the equipotential surfaces for this system (a) out to a distance of about 2.0 m from the origin and (b)far from the origin. In each case, indicate the direction in which the potential increases.

Figure 2027shows a series of equipotentials in a particular region of space, and ve different paths along which an electron is moved. (a) Does the electric eld in this region point to the right, to the left, up, or down? Explain. (b)For each path, indicate whether the work done on the electron by the electric eld is positive, negative, or zero. (c)Rank the paths in order of increasing amount of work done on the electron by the electric eld. Indicate ties where appropriate. (d) Is the electric eld near path Agreater than, less than, or equal to the electric eld near path E? Explain.

Consider a region in space where a uniform electric eld points in the negative xdirection. (a)What is the orientation of the equipotential surfaces? Explain. (b) If you move in the positive x direction, does the electric potential increase or decrease? Explain. (c) What is the distance between the and the equipotentials?

A given system has the equipotential surfaces shown in Figure 2028. (a) What are the magnitude and direction of the electric eld? (b) What is the shortest distance one can move to undergo a change in potential of 5.00 V?

A capacitor is connected to a 9.0-V battery. How much charge is on each plate of the capacitor?

It is desired that of charge be stored on each plate of a capacitor. What potential difference is required between the plates?

To operate a given ash lamp requires a charge of . What capacitance is needed to store this much charge in a capacitor with a potential difference between its plates of 9.0 V?

A parallel-plate capacitor is made from two aluminum-foil sheets, each 6.3 cm wide and 5.4 m long. Between the sheets is a Teon strip of the same width and length that is 0.035 mm thick. What is the capacitance of this capacitor? (The dielectric constant of Teon is 2.1.)

Aparallel-plate capacitor is constructed with circular plates of radius 0.056 m. The plates are separated by 0.25 mm, and the space between the plates is lled with a dielectric with dielectric constant . When the charge on the capacitor is the potential difference between the plates is 750 V. Find the value of the dielectric constant,

A parallel-plate capacitor has plates with an area of and a separation of 0.88 mm. The space between the plates is lled with a dielectric whose dielectric constant is 2.0. (a)What is the potential difference between the plates when the charge on the capacitor plates is ? (b)Will your answer to part (a) increase, decrease, or stay the same if the dielectric constant is increased? Explain. (c)Calculate the potential difference for the case where the dielectric constant is 4.0.

Consider a parallel-plate capacitor constructed from two circular metal plates of radius R. The plates are separated by a distance of 1.5 mm. (a) What radius must the plates have if the capacitance of this capacitor is to be ? (b)If the separation between the plates is increased, should the radius of the plates be increased or decreased to maintain a capacitance of ? Explain.(c)Find the radius of the plates that gives a capacitance of for a plate separation of 3.0 mm.

Aparallel-plate capacitor has plates of area . What plate separation is required if the capacitance is to be 1630 pF? Assume that the space between the plates is lled with (a) air or (b) paper.

(a) Find the magnitude of the charge on each plate when the capacitor is connected to a 12-V battery. (b)Will your answer to part (a) increase, decrease, or stay the same if the separation between the plates is increased? Explain. (c) Calculate the magnitude of the charge on the plates if the separation is 0.90 mm.

Suppose that after walking across a carpeted oor you reach for a doorknob and just before you touch it a spark jumps 0.50 cm from your nger to the knob. Find the minimum voltage needed between your nger and the doorknob to generate this spark.

(a) What plate area is required if an air-lled, parallel-plate capacitor with a plate separation of 2.6 mm is to have a capacitance of 22 pF? (b)What is the maximum voltage that can be applied to this capacitor without causing dielectric breakdown?

Lightning As a crude model for lightning, consider the ground to be one plate of a parallel-plate capacitor and a cloud at an altitude of 550 m to be the other plate. Assume the surface area of the cloud to be the same as the area of a square that is 0.50 km on a side. (a) What is the capacitance of this capacitor? (b) How much charge can the cloud hold before the dielectric strength of the air is exceeded and a spark (lightning) results?

Aparallel-plate capacitor is made from two aluminum-foil sheets, each 3.00 cm wide and 10.0 m long. Between the sheets is a mica strip of the same width and length that is 0.0225 mm thick. What is the maximum charge that can be stored in this capacitor? (The dielectric constant of mica is 5.4, and its dielectric strength is .)

Calculate the work done by a 3.0-V battery as it charges a capacitor in the ash unit of a camera.

DebrillatorAn automatic external debrillator (AED) delivers 125 J of energy at a voltage of 1050 V. What is the capacitance of this device?

Cell MembranesThe membrane of a living cell can be approximated by a parallel-plate capacitor with plates of area , a plate separation of , and a dielectric with a dielectric constant of 4.5. (a)What is the energy stored in such a cell membrane if the potential difference across it is 0.0725 V? (b) Would your answer to part (a) increase, decrease, or stay the same if the thickness of the cell membrane is increased? Explain.

A capacitor is charged by a 1.5-V battery. After being charged, the capacitor is connected to a small electric motor. Assuming 100% efciency, (a) to what height can the motor lift a 5.0-g mass? (b)What initial voltage must the capacitor have if it is to lift a 5.0-g mass through a height of 1.0 cm?

Find the electric energy density between the plates of a parallel-plate capacitor. The potential difference between the plates is 345 V, and the plate separation is 0.223 mm.

What electric eld strength would store 17.5 J of energy in every of space?

An electronic ash unit for a camera contains a capacitor with a capacitance of . When the unit is fully charged and ready for operation, the potential difference between the capacitor plates is 330 V. (a) What is the magnitude of the charge on each plate of the fully charged capacitor? (b)Find the energy stored in the charged-up ash unit.

Aparallel-plate capacitor has plates with an area of and an air-lled gap between the plates that is 2.25 mm thick. The capacitor is charged by a battery to 575 V and then is disconnected from the battery. (a) How much energy is stored in the capacitor? (b) The separation between the plates is now increased to 4.50 mm. How much energy is stored in the capacitor now? (c) How much work is required to increase the separation of the plates from 2.25 mm to 4.50 mm? Explain your reasoning.

Aproton is released from rest in a region of space with a nonzero electric eld. As the proton moves, does it experience an increasing or decreasing electric potential? Explain.

Predict/ExplainAproton is released from rest in a region of space with a nonzero electric eld. (a) As the proton moves, does the electric potential energy of the system increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. As the proton begins to move, its kinetic energy increases. The increase in kinetic energy is equal to the decrease in the electric potential energy of the system. II. Because the proton has a positive charge, its electric potential energy will always increase. III. The proton will move perpendicular to the electric eld, and hence its electric potential energy will remain the same.

In the Bohr model of the hydrogen atom, a proton and an electron are separated by a constant distance r. (a) Would the electric potential energy of the system increase, decrease, or stay the same if the electron is replaced with a proton? Explain. (b) Suppose, instead, that the proton is replaced with an electron. Would the electric potential energy of the system increase, decrease, or stay the same? Explain.

The plates of a parallel-plate capacitor have constant charges of and . Do the following quantities increase, decrease, or remain the same as the separation of the plates is increased? (a) The electric eld between the plates; (b) the potential difference between the plates; (c) the capacitance; (d) the energy stored in the capacitor.

Aparallel-plate capacitor is connected to a battery that maintains a constant potential difference V between the plates. If the plates of the capacitor are pulled farther apart, do the following quantities increase, decrease, or remain the same? (a) The electric eld between the plates; (b)the charge on the plates; (c)the capacitance;(d)the energy stored in the capacitor.

The plates of a parallel-plate capacitor have constant charges of and . Do the following quantities increase, decrease, or remain the same as a dielectric is inserted between the plates? (a) The electric eld between the plates; (b) the potential difference between the plates; (c)the capacitance; (d)the energy stored in the capacitor.

A parallel-plate capacitor is connected to a battery that maintains a constant potential difference V between the plates. If a dielectric is inserted between the plates of the capacitor, do the following quantities increase, decrease, or remain the same? (a) The electric eld between the plates; (b) the charge on the plates;(c)the capacitance; (d)the energy stored in the capacitor

Find the difference in electric potential, , between the points Aand B for the following cases: (a)The electric eld does 0.052 J of work as you move a charge from A to B. (b) The electric eld does of work as you move a charge from Ato B. (c)You perform 0.052 J of work as you slowly move a charge from Ato B.

The separation between the plates of a parallel-plate capacitor is doubled and the area of the plates is halved. How is the capacitance affected?

Aparallel-plate capacitor is connected to a battery that maintains a constant potential difference between the plates. If the spacing between the plates is doubled, how is the magnitude of charge on the plates affected?

Two point charges are placed on the x axis. The charge is at , and the charge is at . (a) There is a point on the xaxis between the two charges where the electric potential is zero. Where is this point? (b) The electric potential also vanishes at a point in one of the following regions: region 1, x between 1.5 m and 5.0 m; region 2, x between and ; region 3, x between and . Identify the appropriate region. (c) Find the value of x referred to in part (b).

A charge of is located at (4.40 m, 6.22 m), and a charge of is located at . What charge must be located at if the electric potential is to be zero at the origin?

The Bohr Model In the Bohr model of the hydrogen atom (see Problem 28) what is the smallest amount of work that must be done on the electron to move it from its circular orbit, with a radius of , to an innite distance from the proton? This value is referred to as the ionization energy of hydrogen.

A charge and a charge are placed at (0.50 m, 0) and , respectively. (a) In Figure 2029, at which of the points A, B, C, or D is the electric potential smallest in value? At which of these points does it have its greatest value? Explain. (b) Calculate the electric potential at points A, B, C, and D.

Repeat Problem 84 for the case where both charges are .

How much work is required to bring three protons, initially innitely far apart, to a conguration where each proton is from the other two? (This is a typical separation for protons in a nucleus.)

A point charge is held xed at the origin. A second point charge, with mass and charge , is placed at the location (0.323 m, 0). (a)Find the electric potential energy of this system of charges. (b)If the second charge is released from rest, what is its speed when it reaches the point (0.121 m, 0)?

Electron Escape SpeedAn electron is at rest just above the surface of a sphere with a radius of 2.7 mm and a uniformly distributed positive charge of . Like a rocket blasting off from the Earth, the electron is given an initial speed radially outward from the sphere. If the electron coasts to innity, where its kinetic energy drops to zero, what is the escape speed,

Quark Model of the Neutron According to the quark model of fundamental particles, neutronsthe neutral particles in an atoms nucleusare composed of three quarks. Two of these quarks are down quarks, each with a charge of ; the third quark is an up quark, with a charge of . This gives the neutron a net charge of zero. What is the electric potential energy of these three quarks, assuming they are equidistant from one another, with a separation distance of ? (Quarks are discussed in Chapter 32.)

Aparallel-plate capacitor is charged to an electric potential of 325 V by moving electrons from one plate to the other. How much work is done in charging the capacitor?

The three charges shown in Figure 2025are held in place as a fourth charge, q, is brought from innity to the point P. The charge q starts at rest at innity and is also at rest when it is placed at the point P. (a) If q is a positive charge, is the work required to bring it to the point P positive, negative, or zero? Explain. (b) Find the value of q if the work needed to bring it to point Pis .

(a)In Figure 2028we see that the electric potential increases by 10.0 V as one moves 4.00 cm in the positive x direction. Use this information to calculate the x component of the electric eld. (Ignore the y direction for the moment.) (b) Apply the same reasoning as in part (a) to calculate the ycomponent of the electric eld. (c) Combine the results from parts (a) and (b) to nd the magnitude and direction of the electric eld for this system.

Electric Catsh The electric catsh (Malapterurus electricus) is an aggressive sh, 1.0 m in length, found today in tropical Africa (and depicted in Egyptian hieroglyphics). The catsh is capable of generating jolts of electricity up to 350 V by producing a positively charged region of muscle near the head and a negatively charged region near the tail. (a) For the same amount of charge, can the catsh generate a higher voltage by separating the charge from one end of its body to the other, as it does, or from one side of the body to the other? Explain. (b)Estimate the charge generated at each end of a catsh as follows: Treat the catsh as a parallel-plate capacitor with plates of area , separation 1.0 m, and lled with a dielectric with a dielectric constant .

As a charge moves along the x axis from to , the electric potential it experiences is shown in Figure 2021. Find the approximate location(s) of the charge when its electric potential energy is (a) and (b) .

Computer KeyboardsMany computer keyboards operate on the principle of capacitance. As shown in Figure 2016, each key forms a small parallel-plate capacitor whose separation is reduced when the key is depressed. (a) Does depressing a key increase or decrease its capacitance? Explain. (b)Suppose the plates for each key have an area of and an initial separation of 0.550 mm. In addition, let the dielectric have a dielectric constant of 3.75. If the circuitry of the computer can detect a change in capacitance of 0.425 pF, what is the minimum distance a key must be depressed to be detected?

A point charge of mass 0.081 kg and charge is suspended by a thread between the vertical parallel plates of a parallel-plate capacitor, as shown in Figure 2030.(a)If the charge deects to the right of vertical, as indicated in the gure, which of the two plates is at the higher electric potential? (b) If the angle of deection is 22, and the separation between the plates is 0.025 m, what is the potential difference between the plates?

Cell Membranes and Dielectrics Many cells in the body have a cell membrane whose inner and outer surfaces carry opposite charges, just like the plates of a parallel-plate capacitor. Suppose a typical cell membrane has a thickness of , and its inner and outer surfaces carry charge densities of and , respectively. In addition, assume that the material in the cell membrane has a dielectric constant of 5.5. (a) Find the direction and magnitude of the electric eld within the cell membrane. (b) Calculate the potential difference between the inner and outer walls of the membrane, and indicate which wall of the membrane has the higher potential.

Long, long ago, on a planet far, far away, a physics experiment was carried out. First, a 0.250-kg ball with zero net charge was dropped from rest at a height of 1.00 m. The ball landed 0.552 s later. Next, the ball was given a net charge of and dropped in the same way from the same height. This time the ball fell for 0.680 s before landing. What is the electric potential at a height of 1.00 m above the ground on this planet, given that the electric potential at ground level is zero? (Air resistance can be ignored.)

Rutherfords Planetary Model of the Atom In 1911, Ernest Rutherford developed a planetary model of the atom, in which a small positively charged nucleus is orbited by electrons. The model was motivated by an experiment carried out by Rutherford and his graduate students, Geiger and Marsden. In this experiment, they red alpha particles with an initial speed of at a thin sheet of gold. (Alpha particles are obtained from certain radioactive decays. They have a charge of and a mass of .) How close can the alpha particles get to a gold nucleus , assuming the nucleus remains stationary? (This calculation sets an upper limit on the size of the gold nucleus. See Chapter 31 for further details.)

(a) One of the charges in Figure 2026 is given an outward kick that sends it off with an initial speed while the other three charges are held at rest. If the moving charge has a mass m, what is its speed when it is innitely far from the other charges? (b) Suppose the remaining charge, which also has a mass m, is now given the same initial speed, . When it is innitely far away from the two charges, is its speed greater than, less than, or the same as the speed found in part (a)? Explain.

Figure 2030 shows a charge with a mass suspended by a thread of length between the plates of a capacitor. (a) Plot the electric potential energy of the system as a function of the angle the thread makes with the vertical. (The electric eld between the plates has a magnitude .) (b) Repeat part (a) for the case of the gravitational potential energy of the system. (c) Show that the total potential energy of the system (electric plus gravitational) is a minimum when the angle satises the equilibrium condition for the charge, . This relation implies that .

The electric potential a distance r from a point charge q is . One meter farther away from the charge the potential is 6140 V. Find the charge q and the initial distance r.

Referring to Problem 84, calculate and plot the electric potential on the circle centered at (0.50 m, 0). Give your results in terms of the angle , dened as follows: is the angle measured counterclockwise from a vertex at the center of the circle, with at point C.

When the potential difference between the plates of a capacitor is increased by 3.25 V, the magnitude of the charge on each plate increases by . What is the capacitance of this capacitor?

The electric potential a distance r from a point charge q is 155 V, and the magnitude of the electric eld is 2240 N/C. Find the values of q and r.

Electric eels produce an electric eld within their body. In which direction does the electric eld point? A. toward the head B. toward the tail C. upward D. downward

As a rough approximation, consider an electric eel to be a parallel-platecapacitor with plates of area separated by 2.0 m and lled with a dielectric whose dielectric constant is . What is the capacitance of the eel in this model? A. B. C. D

In terms of the parallel-plate model of the previous problem, how much charge does an electric eel generate at each end of its body when it produces a voltage of 650 V? A. B. C. D

How much energy is stored by an electric eel when it is charged up to 650 V. Use the same parallel-plate model discussed in the previous two problems. A. B. C. D

Referring to Example 203 Suppose the charge at is replaced with a charge , where . The charge is at the origin. (a) Is the electric potential positive, negative, or zero at the point ? Explain.(b)Find the point between and where the electric potential vanishes. (c) Is there a point in the region where the electric potential passes through zero?

Referring to Example 203 Suppose we can change the location of the charge on the xaxis. The charge (where ) is still at the origin. (a) Where should the charge be placed to ensure that the electric potential vanishes at ? (b) With the location of found in part (a), where does the electric potential pass through zero in the region ?

Referring to Example 203 Suppose the charge at the origin is replaced with a charge , where The charge is still at . (a) Is there a point in the region where the electric potential passes through zero? (b) Find the location between and where the electric potential passes through zero. (c) Find the location in the region where the electric potential passes through zero.