The electric potential everywhere on the xy plane is V 5

In a certain region of space, the electric field is zero. From this fact, what can you conclude about the electric potential in this region? (a) It is zero. (b) It does not vary with position. (c) It is positive. (d) It is negative. (e) None of those answers is necessarily true.

Consider the equipotential surfaces shown in Figure 25.4. In this region of space, what is the approximate direction of the electric field? (a) It is out of the page. (b) It is into the page. (c) It is toward the top of the page. (d) It is toward the bottom of the page. (e) The field is zero

(i) A metallic sphere A of radius 1.00 cm is several centimeters away from a metallic spherical shell B of radius 2.00 cm. Charge 450 nC is placed on A, with no charge on B or anywhere nearby. Next, the two objects are joined by a long, thin, metallic wire (as shown in Fig. 25.19), and finally the wire is removed. How is the charge shared between A and B? (a) 0 on A, 450 nC on B (b) 90.0 nC on A and 360 nC on B, with equal surface charge densities (c) 150 nC on A and 300 nC on B (d) 225 nC on A and 225 nC on B (e) 450 nC on A and 0 on B (ii) A metallic sphere A of radius 1 cm with charge 450 nC hangs on an insulating thread inside an uncharged thin metallic spherical shell B of radius 2 cm. Next, A is made temporarily to touch the inner surface of B. How is the charge then shared between them? Choose from the same possibilities. Arnold Arons, the only physics teacher yet to have his picture on the cover of Time magazine, suggested the idea for this question.

The electric potential at x 5 3.00 m is 120 V, and the electric potential at x 5 5.00 m is 190 V. What is the x component of the electric field in this region, assuming the field is uniform? (a) 140 N/C (b) 2140 N/C (c) 35.0 N/C (d) 235.0 N/C (e) 75.0 N/C

Rank the potential energies of the four systems of particles shown in Figure OQ25.5 from largest to smallest. Include equalities if appropriate.

In a certain region of space, a uniform electric field is in the x direction. A particle with negative charge is carried from x 5 20.0 cm to x 5 60.0 cm. (i) Does the electric potential energy of the chargefield system (a) increase, (b) remain constant, (c) decrease, or (d) change unpredictably? (ii) Has the particle moved to a position where the electric potential is (a) higher than before, (b) unchanged, (c) lower than before, or (d) unpredictable?

Rank the electric potentials at the four points shown in Figure OQ25.7 from largest to smallest.

An electron in an x-ray machine is accelerated through a potential difference of 1.00 3 104 V before it hits the target. What is the kinetic energy of the electron in electron volts? (a) 1.00 3 104 eV (b) 1.60 3 10215 eV (c) 1.60 3 10222 eV (d) 6.25 3 1022 eV (e) 1.60 3 10219 eV

Rank the electric potential energies of the systems of charges shown in Figure OQ25.9 from largest to smallest. Indicate equalities if appropriate.

Four particles are positioned on the rim of a circle. The charges on the particles are 10.500 mC, 11.50 mC, 21.00 mC, and 20.500 mC. If the electric potential at the center of the circle due to the 10.500 mC charge alone is 4.50 3 104 V, what is the total electric potential at the center due to the four charges? (a) 18.0 3 104 V (b) 4.50 3 104V (c) 0 (d) 24.50 3 104 V (e) 9.00 3 104 V

A proton is released from rest at the origin in a uniform electric field in the positive x direction with magnitude 850 N/C. What is the change in the electric potential energy of the protonfield system when the proton travels to x 5 2.50 m? (a) 3.40 3 10216 J (b) 23.40 3 10216 J (c) 2.50 3 10216 J (d) 22.50 3 10216 J (e) 21.60 3 10219 J

A particle with charge 240.0 nC is on the x axis at the point with coordinate x 5 0. A second particle, with charge 220.0 nC, is on the x axis at x 5 0.500 m. (i) Is the point at a finite distance where the electric field is zero (a) to the left of x 5 0, (b) between x 5 0 and x 5 0.500 m, or (c) to the right of x 5 0.500 m? (ii) Is the electric potential zero at this point? (a) No; it is positive. (b) Yes. (c) No; it is negative. (iii) Is there a point at a finite distance where the electric potential is zero? (a) Yes; it is to the left of x 5 0. (b) Yes; it is between x 5 0 and x 5 0.500 m. (c) Yes; it is to the right of x 5 0.500 m. (d) No

A filament running along the x axis from the origin to x 5 80.0 cm carries electric charge with uniform density. At the point P with coordinates (x 5 80.0 cm, y 5 80.0 cm), this filament creates electric potential 100 V. Now we add another filament along the y axis, running from the origin to y 5 80.0 cm, carrying the same amount of charge with the same uniform density. At the same point P, is the electric potential created by the pair of filaments (a) greater than 200 V, (b) 200 V, (c) 100 V, (d) between 0 and 200 V, or (e) 0?

In different experimental trials, an electron, a proton, or a doubly charged oxygen atom (O22), is fired within a vacuum tube. The particles trajectory carries it through a point where the electric potential is 40.0 V and then through a point at a different potential. Rank each of the following cases according to the change in kinetic energy of the particle over this part of its flight from the largest increase to the largest decrease in kinetic energy. In your ranking, display any cases of equality. (a) An electron moves from 40.0 V to 60.0 V. (b) An electron moves from 40.0 V to 20.0 V. (c) A proton moves from 40.0 V to 20.0 V. (d) A proton moves from 40.0 V to 10.0 V. (e) An O22 ion moves from 40.0 V to 60.0 V

A helium nucleus (charge 5 2e, mass 5 6.63 3 10227 kg) traveling at 6.20 3 105 m/s enters an electric field, traveling from point A, at a potential of 1.50 3 103 V, to point B, at 4.00 3 103 V. What is its speed at point B? (a) 7.91 3 105 m/s (b) 3.78 3 105 m/s (c) 2.13 3 105 m/s (d) 2.52 3 106 m/s (e) 3.01 3 108 m/s

Two point charges Q 1 5 15.00 nC and Q 2 5 23.00 nC are separated by 35.0 cm. (a) What is the electric potential at a point midway between the charges? (b) What is the potential energy of the pair of charges? What is the significance of the algebraic sign of your answer?

Two particles, with charges of 20.0 nC and 220.0 nC, are placed at the points with coordinates (0, 4.00 cm) and (0, 24.00 cm) as shown in Figure P25.17. A particle with charge 10.0 nC is located at the origin. (a) Find the electric potential energy of the configuration of the three fixed charges. (b) A fourth particle, with a mass of 2.00 3 10213 kg and a charge of 40.0 nC, is released from rest at the point (3.00 cm, 0). Find its speed after it has moved freely to a very large distance away.

The two charges in Figure P25.18 are separated by a distance d 5 2.00 cm, and Q 5 15.00 nC. Find (a) the electric potential at A, (b) the electric potential at B, and (c) the electric potential difference between B and A.

Given two particles with 2.00-mC charges as shown in Figure P25.19 and a particle with charge q 5 1.28 3 10218 C at the origin, (a) what is the net force exerted by the two 2.00-mC charges on the charge q? (b) What is the electric field at the origin due to the two 2.00-mC particles? (c)What is the electric potential at the origin due to the two 2.00-mC particles?

At a certain distance from a charged particle, the magnitude of the electric field is 500 V/m and the electric potential is 23.00 kV. (a) What is the distance to the particle? (b) What is the magnitude of the charge?

Four point charges each having charge Q are located at the corners of a square having sides of length a. Find expressions for (a) the total electric potential at the center of the square due to the four charges and (b) the work required to bring a fifth charge q from infinity to the center of the square.

The three charged particles in Figure P25.22 are at the vertices of an isosceles triangle (where d 5 2.00 cm). Taking q 5 7.00 mC, calculate the electric potential at point A, the midpoint of the base.

A particle with charge 1q is at the origin. A particle with charge 22q is at x 5 2.00 m on the x axis. (a) For what finite value(s) of x is the electric field zero? (b) For what finite value(s) of x is the electric potential zero?

Show that the amount of work required to assemble four identical charged particles of magnitude Q at the corners of a square of side s is 5.41keQ 2/s

Two particles each with charge 12.00 mC are located on the x axis. One is at x 5 1.00 m, and the other is at x 5 21.00 m. (a) Determine the electric potential on the y axis at y 5 0.500 m. (b) Calculate the change in electric potential energy of the system as a third charged particle of 23.00 mC is brought from infinitely far away to a position on the y axis at y 5 0.500 m.

Two charged particles of equal magnitude are located along the y axis equal distances above and below the x axis as shown in Figure P25.26. (a) Plot a graph of the electric potential at points along the x axis over the interval 23a , x , 3a. You should plot the potential in units of ke Q /a. (b) Let the charge of the particle located at y 5 2a be negative. Plot the potential along the y axis over the interval 24a , y , 4a.

Four identical charged particles (q 5 110.0 mC) are located on the corners of a rectangle as shown in Figure P25.27. The dimensions of the rectangle are L 5 60.0 cm and W 5 15.0 cm. Calculate the change in electric potential energy of the system as the particle at the lower left corner in Figure P25.27 is brought to this position from infinitely far away. Assume the other three particles in Figure P25.27 remain fixed in position

Three particles with equal positive charges q are at the corners of an equilateral triangle of side a as shown in Figure P25.28. (a) At what point, if any, in the plane of the particles is the electric potential zero? (b) What is the electric potential at the position of one of the particles due to the other two particles in the triangle?

Five particles with equal negative charges 2q are placed symmetrically around a circle of radius R. Calculate the electric potential at the center of the circle.

Review. A light, unstressed spring has length d. Two identical particles, each with charge q, are connected to the opposite ends of the spring. The particles are held stationary a distance d apart and then released at the same moment. The system then oscillates on a frictionless, horizontal table. The spring has a bit of internal kinetic friction, so the oscillation is damped. The particles eventually stop vibrating when the distance between them is 3d. Assume the system of the spring and two charged particles is isolated. Find the increase in internal energy that appears in the spring during the oscillations.

Review. Two insulating spheres have radii 0.300 cm and 0.500 cm, masses 0.100 kg and 0.700 kg, and uniformly distributed charges 22.00 mC and 3.00 mC. They are released from rest when their centers are separated by 1.00 m. (a) How fast will each be moving when they collide? (b) What If? If the spheres were conductors, would the speeds be greater or less than those calculated in part (a)? Explain.

Review. Two insulating spheres have radii r1 and r2, masses m 1 and m 2, and uniformly distributed charges 2q1 and q2. They are released from rest when their centers are separated by a distance d. (a) How fast is each moving when they collide? (b) What If? If the spheres were conductors, would their speeds be greater or less than those calculated in part (a)? Explain.

How much work is required to assemble eight identical charged particles, each of magnitude q, at the corners of a cube of side s?

Four identical particles, each having charge q and mass m, are released from rest at the vertices of a square of side L. How fast is each particle moving when their distance from the center of the square doubles?

In 1911, Ernest Rutherford and his assistants Geiger and Marsden conducted an experiment in which they scattered alpha particles (nuclei of helium atoms) from thin sheets of gold. An alpha particle, having charge 12e and mass 6.64 3 10227 kg, is a product of certain radioactive decays. The results of the experiment led Rutherford to the idea that most of an atoms mass is in a very small nucleus, with electrons in orbit around it. (This is the planetary model of the atom, which well study in Chapter 42.) Assume an alpha particle, initially very far from a stationary gold nucleus, is fired with a velocity of 2.00 3 107 m/s directly toward the nucleus (charge 179e). What is the smallest distance between the alpha particle and the nucleus before the alpha particle reverses direction? Assume the gold nucleus remains stationary.

Figure P25.36 represents a graph of the electric potential in a region of space versus position x, where the electric field is parallel to the x axis. Draw a graph of the x component of the electric field versus x in this region.

The potential in a region between x 5 0 and x 5 6.00 m is V 5 a 1 bx, where a 5 10.0 V and b 5 27.00 V/m. Determine (a) the potential at x 5 0, 3.00 m, and 6.00 m and (b) the magnitude and direction of the electric field at x 5 0, 3.00 m, and 6.00 m.

An electric field in a region of space is parallel to the x axis. The electric potential varies with position as shown in Figure P25.38. Graph the x component of the electric field versus position in this region of space.

Over a certain region of space, the electric potential is V 5 5x 2 3x 2y 1 2yz 2. (a) Find the expressions for the x, y, and z components of the electric field over this region. (b) What is the magnitude of the field at the point P that has coordinates (1.00, 0, 22.00) m?

Figure P25.40 shows several equipotential lines, each labeled by its potential in volts. The distance between the lines of the square grid represents 1.00 cm. (a) Is the magnitude of the field larger at A or at B ? Explain how you can tell. (b) Explain what you can determine about E S at B. (c)Represent what the electric field looks like by drawing at least eight field lines.

The electric potential inside a charged spherical conductor of radius R is given by V 5 keQ /R, and the potential outside is given by V 5 keQ /r. Using Er 5 2dV/dr, derive the electric field (a) inside and (b) outside this charge distribution.

It is shown in Example 25.7 that the potential at a point P a distance a above one end of a uniformly charged rod of length , lying along the x axis is V 5 ke Q , ln a , 1 "a 2 1 ,2 a b Use this result to derive an expression for the y component of the electric field at P.

Consider a ring of radius R with the total charge Q spread uniformly over its perimeter. What is the potential difference between the point at the center of the ring and a point on its axis a distance 2R from the center?

A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle as shown in Figure P25.44. The rod has a total charge of 27.50 mC. Find the electric potential at O, the center of the semicircle.

A rod of length L (Fig. P25.45) lies along the x axis with its left end at the origin. It has a nonuniform charge density l 5 ax, where a is a positive constant. (a) What are the units of a? (b) Calculate the electric potential atA.

For the arrangement described in Problem 45, calculate the electric potential at point B, which lies on the perpendicular bisector of the rod a distance b above the x axis.

A wire having a uniform linear charge density l is bent into the shape shown in Figure P25.47. Find the electric potential at point O.

The electric field magnitude on the surface of an irregularly shaped conductor varies from 56.0 kN/C to 28.0 kN/C. Can you evaluate the electric potential on the conductor? If so, find its value. If not, explain why not.

How many electrons should be removed from an initially uncharged spherical conductor of radius 0.300 m to produce a potential of 7.50 kV at the surface?

A spherical conductor has a radius of 14.0 cm and a charge of 26.0 mC. Calculate the electric field and the electric potential at (a) r 5 10.0 cm, (b) r 5 20.0 cm, and (c) r 5 14.0 cm from the center.

Electric charge can accumulate on an airplane in flight. You may have observed needle-shaped metal extensions on the wing tips and tail of an airplane. Their purpose is to allow charge to leak off before much of it accumulates. The electric field around the needle is much larger than the field around the body of the airplane and can become large enough to produce dielectric breakdown of the air, discharging the airplane. To model this process, assume two charged spherical conductors are connected by a long conducting wire and a 1.20-mC charge is placed on the combination. One sphere, representing the body of the airplane, has a radius of 6.00 cm; the other, representing the tip of the needle, has a radius of 2.00 cm. (a) What is the electric potential of each sphere? (b) What is the electric field at the surface of each sphere?

Lightning can be studied with a Van de Graaff generator, which consists of a spherical dome on which charge is continuously deposited by a moving belt. Charge can be added until the electric field at the surface of the dome becomes equal to the dielectric strength of air. Any more charge leaks off in sparks as shown in Figure P25.52. Assume the dome has a diameter of 30.0 cm and is surrounded by dry air with a breakdown electric field of 3.00 3 106 V/m. (a) What is the maximum potential of the dome? (b) What is the maximum charge on the dome?

Why is the following situation impossible? In the Bohr model of the hydrogen atom, an electron moves in a circular orbit about a proton. The model states that the electron can exist only in certain allowed orbits around the proton: those whose radius r satisfies r 5 n2 (0.052 9 nm), where n 5 1, 2, 3, . . . . For one of the possible allowed states of the atom, the electric potential energy of the system is 213.6 eV.

Review. In fair weather, the electric field in the air at a particular location immediately above the Earths surface is 120 N/C directed downward. (a) What is the surface charge density on the ground? Is it positive or negative? (b) Imagine the surface charge density is uniform over the planet. What then is the charge of the whole surface of the Earth? (c) What is the Earths electric potential due to this charge? (d) What is the difference in potential between the head and the feet of a person 1.75 m tall? (Ignore any charges in the atmosphere.) (e) Imagine the Moon, with 27.3% of the radius of the Earth, had a charge 27.3% as large, with the same sign. Find the electric force the Earth would then exert on the Moon. (f) State how the answer to part (e) compares with the gravitational force the Earth exerts on the Moon

Review. From a large distance away, a particle of mass 2.00 g and charge 15.0 mC is fired at 21.0i ^ m/s straight toward a second particle, originally stationary but free to move, with mass 5.00 g and charge 8.50 mC. Both particles are constrained to move only along the x axis. (a) At the instant of closest approach, both particles will be moving at the same velocity. Find this velocity. (b) Find the distance of closest approach. After the interaction, the particles will move far apart again. At this time, find the velocity of (c) the 2.00-g particle and (d) the 5.00-g particle.

eview. From a large distance away, a particle of mass m1 and positive charge q1 is fired at speed v in the positive x direction straight toward a second particle, originally stationary but free to move, with mass m2 and positive charge q2. Both particles are constrained to move only along the x axis. (a) At the instant of closest approach, both particles will be moving at the same velocity. Find this velocity. (b) Find the distance of closest approach. After the interaction, the particles will move far apart again. At this time, find the velocity of (c) the particle of mass m1 and (d) the particle of mass m2.

The liquid-drop model of the atomic nucleus suggests high-energy oscillations of certain nuclei can split the nucleus into two unequal fragments plus a few neutrons. The fission products acquire kinetic energy from their mutual Coulomb repulsion. Assume the charge is distributed uniformly throughout the volume of each spherical fragment and, immediately before separating, each fragment is at rest and their surfaces are in contact. The electrons surrounding the nucleus can be ignored. Calculate the electric potential energy (in electron volts) of two spherical fragments from a uranium nucleus having the following charges and radii: 38e and 5.50 3 10215 m, and 54e and 6.20 3 10215 m.

On a dry winter day, you scuff your leather-soled shoes across a carpet and get a shock when you extend the tip of one finger toward a metal doorknob. In a dark room, you see a spark perhaps 5 mm long. Make orderof-magnitude estimates of (a) your electric potential and (b) the charge on your body before you touch the doorknob. Explain your reasoning.

The electric potential immediately outside a charged conducting sphere is 200 V, and 10.0 cm farther from the center of the sphere the potential is 150 V. Determine (a) the radius of the sphere and (b) the charge on it. The electric potential immediately outside another charged conducting sphere is 210 V, and 10.0 cm farther from the center the magnitude of the electric field is 400 V/m. Determine (c) the radius of the sphere and (d) its charge on it. (e) Are the answers to parts (c) and (d) unique?

(a) Use the exact result from Example 25.4 to find the electric potential created by the dipole described in the example at the point (3a, 0). (b) Explain how this answer compares with the result of the approximate expression that is valid when x is much greater than a.

Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius R 5 0.100 m to a total charge Q 5 125 mC.

Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius R to a total charge Q.

The electric potential everywhere on the xy plane is V 5 36 "1x 1 12 2 1 y 2 2 45 "x 2 1 1y 2 22 2 where V is in volts and x and y are in meters. Determine the position and charge on each of the particles that create this potential.

Why is the following situation impossible? You set up an apparatus in your laboratory as follows. The x axis is the symmetry axis of a stationary, uniformly charged ring of radius R 5 0.500 m and charge Q 5 50.0 mC (Fig. P25.64). You place a particle with charge Q 5 50.0 mC and mass m 5 0.100kg at the center of the ring and arrange for it to be constrained to move only along the x axis. When it is displaced slightly, the particle is repelled by the ring and accelerates along the x axis. The particle moves faster than you expected and strikes the opposite wall of your laboratory at 40.0 m/s.

From Gausss law, the electric field set up by a uniform line of charge is E S 5 a l 2pP0r b r^ where r^ is a unit vector pointing radially away from the line and l is the linear charge density along the line. Derive an expression for the potential difference between r 5 r1 and r 5 r2.

A uniformly charged filament lies along the x axis between x 5 a 5 1.00 m and x 5 a 1 , 5 3.00 m as shown in Figure P25.66. The total charge on the filament is 1.60 nC. Calculate successive approximations for the electric potential at the origin by modeling the filament as (a) a single charged particle at x 5 2.00 m, (b) two 0.800-nC charged particles at x 5 1.5 m and x 5 2.5 m, and (c) four 0.400-nC charged particles at x 5 1.25 m, x 5 1.75 m, x 5 2.25 m, and x 5 2.75 m. (d) Explain how the results compare with the potential given by the exact expression V 5 keQ , ln a , 1 a a b

The thin, uniformly charged rod shown in Figure P25.67 has a linear charge density l. Find an expression for the electric potential at P

A GeigerMueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius ra and a coaxial cylindrical wire (the anode) of radius rb (Fig. P25.68a). The charge per unit length on the anode is l, and the charge per unit length on the cathode is 2l. A gas fills the space between the electrodes. When the tube is in use (Fig. P25.68b) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is DV 5 2ke l ln a ra rb b (b) Show that the magnitude of the electric field in the space between cathode and anode is E 5 DV ln 1ra/rb 2 a 1 r b where r is the distance from the axis of the anode to the point where the field is to be calculated.

Review. Two parallel plates having charges of equal magnitude but opposite sign are separated by 12.0 cm. Each plate has a surface charge density of 36.0 nC/m2. A proton is released from rest at the positive plate. Determine (a) the magnitude of the electric field between the plates from the charge density, (b) the potential difference between the plates, (c) the kinetic energy of the proton when it reaches the negative plate, (d) the speed of the proton just before it strikes the negative plate, (e) the acceleration of the proton, and (f) the force on the proton. (g) From the force, find the magnitude of the electric field. (h) How does your value of the electric field compare with that found in part (a)?

When an uncharged conducting sphere of radius a is placed at the origin of an xyz coordinate system that lies in an initially uniform electric field E S 5 E0 k^ , the resulting electric potential is V(x, y, z) 5 V0 for points inside the sphere and V 1x, y, z2 5 V0 2 E0 z 1 E0a3 z 1x 2 1 y 2 1 z 2 2 3/2 for points outside the sphere, where V0 is the (constant) electric potential on the conductor. Use this equation to determine the x, y, and z components of the resulting electric field (a) inside the sphere and (b) outside the sphere.

An electric dipole is located along the y axis as shown in Figure P25.71. The magnitude of its electric dipole moment is defined as p 5 2aq. (a) At a point P, which is far from the dipole (r .. a), show that the electric potential is V 5 ke p cos u r 2 (b) Calculate the radial component Er and the perpendicular component E u of the associated electric field. Note that E u 5 2(1/r)('V/'u). Do these results seem reasonable for (c) u 5 908 and 08? (d) For r 5 0? (e) For the dipole arrangement shown in Figure P25.71, express V in terms of Cartesian coordinates using r 5 (x 2 1 y 2)1/2 and cos u 5 y 1x 2 1 y 2 2 1/2 (f) Using these results and again taking r .. a, calculate the field components Ex and Ey.

A solid sphere of radius R has a uniform charge density r and total charge Q. Derive an expression for its total electric potential energy. Suggestion: Imagine the sphere is constructed by adding successive layers of concentric shells of charge dq 5 (4pr 2 dr)r and use dU 5 V dq.

A disk of radius R (Fig. P25.73) has a nonuniform surface charge density s 5 Cr, where C is a constant and r is measured from the center of the disk to a point on the surface of the disk. Find (by direct integration) the electric potential at P.

Four balls, each with mass m, are connected by four nonconducting strings to form a square with side a as shown in Figure P25.74. The assembly is placed on a nonconducting, frictionless, horizontal surface. Balls 1 and 2 each have charge q, and balls 3 and 4 are uncharged. After the string connecting balls 1 and 2 is cut, what is the maximum speed of balls 3 and 4?

(a) A uniformly charged cylindrical shell with no end caps has total charge Q , radius R, and length h. Determine the electric potential at a point a distance d from the right end of the cylinder as shown in Figure P25.75. Suggestion: Use the result of Example 25.5 by treating the cylinder as a collection of ring charges. (b) What If? Use the result of Example 25.6 to solve the same problem for a solid cylinder

As shown in Figure P25.76, two large, parallel, vertical conducting plates separated by distance d are charged so that their potentials are 1V0 and 2V0. A small conducting ball of mass m and radius R (where R ,, d) hangs midway between the plates. The thread of length L supporting the ball is a conducting wire connected to ground, so the potential of the ball is fixed at V 5 0. The ball hangs straight down in stable equilibrium when V0 is sufficiently small. Show that the equilibrium of the ball is unstable if V0 exceeds the critical value 3ked 2 mg/14RL24 1/2. Suggestion: Consider the forces on the ball when it is displaced a distance x ,, L.

A particle with charge q is located at x 5 2R, and a particle with charge 22q is located at the origin. Prove that the equipotential surface that has zero potential is a sphere centered at (24R/3, 0, 0) and having a radius r 5 2 3R.