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Master the Concepts

Electric potential energy can be stored in an electric field. The electric potential energy of two point charges separated by a distance

*r*is(17-1)The signs of

*q*_{1}and*q*_{2}determine whether the electric potential energy is positive or negative. For more than two charges, the electric potential energy is the scalar sum of the individual potential energies for each*pair*of charges.The electric potential

*V*at a point is the electric potential energy per unit charge:(17-3)In Eq. (17-3),

*U*_{E}is the electric potential energy due to the interaction of a moveable charge*q*with a collection of fixed charges and*V*is the electric potential due to that collection of fixed charges. Both*U*_{E}and*V*are functions of the position of the moveable charge*q.*Electric potential, like electric potential energy, is a scalar quantity. The SI unit for potential is the volt (1 V = 1 J/C).

If a point charge

*q*moves through a potential difference Δ*V,*then the change in electric potential energy is(17-7)The electric potential at a distance

*r*from a point charge*Q*is(17-8)The potential at a point

*P*due to*N*point charges is the sum of the potentials due to each charge.An equipotential surface has the same potential at every point on the surface. An equipotential surface is perpendicular to the electric field at all points. No change in electric potential energy occurs when a charge moves from one position to another on an equipotential surface. If equipotential surfaces are drawn such that the potential difference between adjacent surfaces is constant, then the surfaces are closer together where the field is stronger.

The electric field always points in the direction of maximum potential decrease.

The potential difference that occurs when you move a distance

*d*in the direction of a uniform electric field of magnitude*E*is(17-10)The electric field has units of

(17-11)- Page 629
In electrostatic equilibrium, every point in a conductor must be at the same potential.

A capacitor consists of two conductors (the

*plates*) that are given opposite charges. A capacitor stores charge and electric potential energy. Capacitance is the ratio of the magnitude of charge on each plate (*Q*) to the electric potential difference between the plates (Δ*V*). Capacitance is measured in farads (F).(17-14)The capacitance of a parallel plate capacitor is

(17-16)where

*A*is the area of each plate,*d*is their separation, and ε_{0}is the permittivity of free space [ε_{0}= 1/(4π*k*) = 8.854 × 10^{-12}C^{2}/(*N*· m^{2})]. If vacuum separates the plates, κ = 1; otherwise, κ > 1 is the dielectric constant of the dielectric (the insulating material). If a dielectric is immersed in an external electric field, the dielectric constant is the ratio of the external electric field*E*_{0}to the electric field*E*in the dielectric.(17-17)The dielectric constant is a measure of the ease with which the insulating material can be polarized.

The dielectric strength is the electric field strength at which dielectric breakdown occurs and the material becomes a conductor.

The energy stored in a capacitor is

(17-18)The energy density

*u*—the electric potential energy per unit volume—associated with an electric field is(17-19)

Conceptual Questions

A negatively charged particle with charge -

*q*is far away from a positive charge +*Q*that is fixed in place. As -*q*moves closer to +*Q,*(a) does the electric field do positive or negative work? (b) Does -*q*move through a potential increase or a potential decrease? (c) Does the electric potential energy increase or decrease? (d) Repeat questions (a)–(c) if the fixed charge is instead negative (-*Q*).Dry air breaks down for a voltage of about 3000 V/mm. Is it possible to build a parallel plate capacitor with a plate spacing of 1 mm that can be charged to a potential difference greater than 3000 V? If so, explain how.

A bird is perched on a high-voltage power line whose potential varies between −100 kV and +100 kV. Why is the bird not electrocuted?

A positive charge is initially at rest in an electric field and is free to move. Does the charge start to move toward a position of higher or lower potential? What happens to a negative charge in the same situation?

Points

*A*and*B*are at the same potential. What is the total work that must be done by an external agent to move a charge from*A*to*B?*Does your answer mean that no external force need be applied? Explain.A point charge moves to a region of higher potential and yet the electric potential energy

*decreases.*How is this possible?Why are all parts of a conductor at the same potential in electrostatic equilibrium?

If

*E*= 0 at a single point, then a point charge placed at that point will feel no electric force. What does it mean if the*potential*is zero at a point? Are there any assumptions behind your answer?If

*E*= 0 everywhere throughout a region of space, what do we know is true about the potential at points in that region?Explain why the woman's hair in Fig. 17.13 stands on end. Why are the hairs directed approximately radially away from her scalp? [

*Hint:*Think of her head as a conducting sphere.]If the potential is the same at every point throughout a region of space, is the electric field the same at every point in that region? What can you say about the magnitude of in the region? Explain.

If a uniform electric field exists in a region of space, is the potential the same at all points in the region? Explain.

When we talk about the potential difference between the plates of a capacitor, shouldn't we really specify two

*points,*one on each plate, and talk about the potential difference between those points? Or doesn't it matter which points we choose? Explain.A swimming pool is filled with water (total mass

*M*) to a height*h.*Explain why the gravitational potential energy of the water (taking*U*= 0 at ground level) is . Where does the factor of come from? How much work must be done to fill the pool, if there is a ready supply of water at ground level? What does this have to do with capacitors? [*Hint:*Make an analogy between the capacitor and the pool. What is analogous to the water? What quantity is analogous to*M?*What quantity is analogous to*gh?*]- Page 630
The charge on a capacitor doubles. What happens to its capacitance?

During a thunderstorm, some cows gather under a large tree. One cow stands facing directly toward the tree. Another cow stands at about the same distance from the tree, but it faces sideways (tangent to a circle centered on the tree). Which cow do you think is more likely to be killed if lightning strikes the tree? [

*Hint:*Think about the potential difference between the cows' front and hind legs in the two positions.]If we know the potential at a single point, what (if anything) can we say about the magnitude of the electric field at that same point?

In Fig. 17.13, why is the person touching the dome of the van de Graaff generator not electrocuted even though there may be a potential difference of hundreds of thousands of volts between her and the ground?

The electric field just above Earth's surface on a clear day in an open field is about 150 V/m downward. Which is at a higher potential: the Earth or the upper atmosphere?

A parallel plate capacitor has the space between the plates filled with a slab of dielectric with κ = 3. While the capacitor is connected to a battery, the dielectric slab is removed. Describe

*quantitatively*what happens to the capacitance, the potential difference, the charge on the plates, the electric field, and the energy stored in the capacitor as the slab is removed. [*Hint:*First figure out which quantities remain constant.]Repeat Question 20 if the capacitor is charged and then disconnected from the battery before removing the dielectric slab.

A charged parallel plate capacitor has the space between the plates filled with air. The capacitor has been disconnected from the battery that charged it. Describe

*quantitatively*what happens to the capacitance, the potential difference, the charge on the plates, the electric field, and the energy stored in the capacitor as the plates are moved closer together. [*Hint:*First figure out which quantities remain constant.]A positive charge +2μC and a negative charge −5 μC lie on a line. In which region or regions (

*A,**B,**C*) is there a point on the line a finite distance away where the potential is zero? Explain your reasoning. Are there any points where both the electric field and the potential are zero?

Multiple-Choice Questions

In all problems, we assign the potential due to a point charge to be zero at an infinite distance from the charge.

Two charges are located at opposite corners (

*A*and*C*) of a square. We do not know the magnitude or sign of these charges. What can be said about the potential at corner*B*relative to the potential at corner*D?*(a) It is the same as that at

*D.*(b) It is different from that at

*D.*(c) It is the same as that at

*D*only if the charges at*A*and*C*are equal.(d) It is the same as that at

*D*only if the charges at*A*and*C*are equal in magnitude and opposite in sign.Among these choices, which is/are correct units for electric field?

(a) N/kg only

(b) N/C only

(c) N only

(d) N · m/C only

(e) V/m only

(f) both N/C and V/m

In the diagram, the potential is zero at which of the points

*A*–*E?*(a)

*B,**D,*and*E*(b)

*B*only(c)

*A,**B,*and*C*(d) all five points

(e) all except

*B*Which of these units can be used to measure electric potential?

(a) N/C

(b) J

(c) V · m

(d) V/m

(e)

A parallel plate capacitor is attached to a battery that supplies a constant potential difference. While the battery is still attached, the parallel plates are separated a little more. Which statement describes what happens?

(a) The electric field increases and the charge on the plates decreases.

(b) The electric field remains constant and the charge on the plates increases.

Page 631(c) The electric field remains constant and the charge on the plates decreases.

(d) Both the electric field and the charge on the plates decrease.

A capacitor has been charged with +

*Q*on one plate and -*Q*on the other plate. Which of these statements is true?(a) The potential difference between the plates is

*QC.*(b) The energy stored is

(c) The energy stored is

(d) The potential difference across the plates is

*Q*^{2}/(2*C*).(e) None of the previous statements is true.

Two solid metal spheres of different radii are far apart. The spheres are connected by a fine metal wire. Some charge is placed on one of the spheres. After electrostatic equilibrium is reached, the wire is removed. Which of these quantities will be the

*same*for the two spheres?(a) the charge on each sphere

(b) the electric field inside each sphere, at the same distance from the center of the spheres

(c) the electric field just outside the surface of each sphere

(d) the electric potential at the surface of each sphere

(e) both (b) and (c) (f) both (b) and (d)

(g) both (a) and (c)

A large negative charge -

*Q*is located in the vicinity of points*A*and*B.*Suppose a positive charge +*q*is moved at constant speed from*A*to*B*by an external agent. Along which of the paths shown in the figure will the work done by the field be the greatest?(a) path 1 (b) path 2 (c) path 3 (d) path 4

(e) Work is the same along all four paths.

A tiny charged pellet of mass

*m*is suspended at rest between two horizontal, charged metallic plates. The lower plate has a positive charge and the upper plate has a negative charge. Which statement in the answers here is*not*true?(a) The electric field between the plates points vertically upward.

(b) The pellet is negatively charged.

(c) The magnitude of the electric force on the pellet is equal to

*mg.*(d) The plates are at different potentials.

Two positive 2.0-μC point charges are placed as shown in part (a) of the figure. The distance from each charge to the point

*P*is 0.040 m. Then the charges are rearranged as shown in part (b) of the figure. Which statement is now true concerning and*V*at point*P?*(a) The electric field and the electric potential are both zero.

(b) , but

*V*is the same as before the charges were moved.(c)

*V*= 0, but is the same as before the charges were moved.(d) is the same as before the charges were moved, but

*V*is less than before.(e) Both and

*V*have changed and neither is zero.In the diagram, which two points are closest to being at the same potential?

(a)

*A*and*D*(b)

*B*and*C*(c)

*B*and*D*(d)

*A*and*C*In the diagram, which point is at the lowest potential?

(a)

*A*(b)

*B*(c)

*C*(d)

*D*

Problems

Combination conceptual/quantitative problem

Biological or medical application

Challenging problem

Detailed solution in the Student Solutions Manual

Problems paired by concept

Interactive or tutorial

17.1 Electric Potential Energy

**1.**Two point charges, +5.0 μC and -2.0 μC, are separated by 5.0 m. What is the electric potential energy?2. A hydrogen atom has a single proton at its center and a single electron at a distance of approximately 0.0529 nm from the proton. (a) What is the electric potential energy in joules? (b) What is the significance of the sign of the answer?

3. How much work is done by an applied force that moves two charges of 6.5 μC that are initially very far apart to a distance of 4.5 cm apart?

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4. The nucleus of a helium atom contains two protons that are approximately 1 fm apart. How much work must be done by an external agent to bring the two protons from an infinite separation to a separation of 1.0 fm?

**5.**How much work does it take for an external force to set up the arrangement of charged objects in the diagram on the corners of a right triangle when the three objects are initially very far away from each other?

**Problems 6–9.** Two point charges (+10.0 nC and -10.0 nC) are located 8.00 cm apart. For each problem, let *U* = 0 when *all* of the charges are separated by infinite distances.

6. What is the potential energy for these two charges?

**7.**What is the potential energy if a third point charge*q*= -4.2 nC is placed at point*a?*8. What is the potential energy if a third point charge

*q*= -4.2 nC is placed at point*b?***9.**What is the potential energy if a third point charge*q*= -4.2 nC is placed at point*c?*

10. Find the electric potential energy for the following array of charges: charge

*q*_{1}= +4.0 μC is located at (*x,**y*) = (0.0, 0.0) m; charge*q*_{2}= +3.0 μC is located at (4.0, 3.0) m; and charge*q*_{3}= -1.0 μC is located at (0.0, 3.0) m.

**11.**In the diagram, how much work is done*by the electric field*as a third charge*q*_{3}= +2.00 nC is moved from infinity to point*a?*12. In the diagram, how much work is done

*by the electric field*as a third charge*q*_{3}= +2.00 nC is moved from infinity to point*b?***13.**In the diagram, how much work is done*by the electric field*as a third charge*q*_{3}= +2.00 nC is moved from point*a*to point*b?*14. In the diagram, how much work is done

*by the electric field*as a third charge*q*_{3}= +2.00 nC is moved from point*b*to point*c?*

17.2 Electric Potential

15. A point charge

*q*= +3.0 nC moves through a potential difference Δ*V*=*V*_{f}-*V*_{i}= +25 V. What is the change in the electric potential energy?16. An electron is moved from point

*A,*where the electric potential is*V*_{A}= −240 V, to point*B,*where the electric potential is*V*_{B}= −360 V. What is the change in the electric potential energy?

**17.**Find the electric field and the potential at the center of a square of side 2.0 cm with charges of +9.0 μC at each corner.18. Find the electric field and the potential at the center of a square of side 2.0 cm with a charge of +9.0 μC at one corner of the square and with charges of -3.0 μC at the remaining three corners.

19. A charge

*Q*= -50.0 nC is located 0.30 m from point*A*and 0.50 m from point*B.*(a) What is the potential at*A?*(b) What is the potential at*B?*(c) If a point charge*q*is moved from*A*to*B*while*Q*is fixed in place, through what potential difference does it move? Does its potential increase or decrease? (d) If*q*= -1.0 nC, what is the change in electric potential energy as it moves from*A*to*B?*Does the potential energy increase or decrease? (e) How much work is done by the electric field due to charge*Q*as*q*moves from*A*to*B?*

20. A charge of +2.0 mC is located at

*x*= 0,*y*= 0 and a charge of -4.0 mC is located at*x*= 0,*y*= 3.0 m. What is the electric potential due to these charges at a point with coordinates*x*= 4.0 m,*y*= 0?**21.**The electric potential at a distance of 20.0 cm from a point charge is +1.0 kV (assuming*V*= 0 at infinity). (a) Is the point charge positive or negative? (b) At what distance is the potential +2.0 kV?Tutorial: Field and Potential of Point Charge

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22. A spherical conductor with a radius of 75.0 cm has an electric field of magnitude 8.40 × 10

^{5}V/m just outside its surface. What is the electric potential just outside the surface, assuming the potential is zero far away from the conductor? 23. An array of four charges is arranged along the

*x*-axis at intervals of 1.0 m. (a) If two of the charges are +1.0 μC and two are -1.0 μC, draw a configuration of these charges that minimizes the potential at*x*= 0. (b) If three of the charges are the same,*q*= +1.0 μC, and the charge at the far right is -1.0 μC, what is the potential at the origin?24. At a point

*P,*a distance*R*_{0}from a positive charge*Q*_{0}, the electric field has a magnitude*E*_{0}= 100 N/C and the electric potential is*V*_{0}= 10 V. The charge is now increased by a factor of three, becoming 3*Q*_{0}. (a) At what distance,*R*_{E}, from the charge 3*Q*_{0}will the electric field have the same value,*E*=*E*_{0}; and (b) at what distance,*R*_{V}, from the charge 3*Q*_{0}will the electric potential have the same value,*V*=*V*_{0}?**25.**Charges of +2.0 nC and -1.0 nC are located at opposite corners,*A*and*C,*respectively, of a square which is 1.0 m on a side. What is the electric potential at a third corner,*B,*of the square (where there is no charge)?26. (a) Find the electric potential at points

*a*and*b*for charges of +4.2 nC and -6.4 nC located as shown in the figure. (b) What is the potential difference Δ*V*for a trip from*a*to*b?*(c) How much work must be done by an external agent to move a point charge of +1.50 nC from*a*to*b?*27. (a) Find the potential at points

*a*and*b*in the diagram for charges*Q*_{1}= +2.50 nC and*q*_{2}= -2.50 nC. (b) How much work must be done by an external agent to bring a point charge*q*from infinity to point*b?*28. (a) In the diagram, what are the potentials at points

*a*and*b?*Let*V*= 0 at infinity. (b) What is the change in electric potential energy if a third charge*q*_{3}= +2.00 nC is moved from point*a*to point*b?*(If you have done Problem 13, compare your answers.)**29.**(a) In the diagram, what are the potentials at points*b*and*c?*Let*V*= 0 at infinity. (b) What is the change in electric potential energy if a third charge*q*_{3}= +2.00 nC is moved from point*b*to point*c?*(If you have done Problem 14, compare your answers.)

17.3 The Relationship Between Electric Field and Potential

30. By rewriting each unit in terms of kilograms, meters, seconds, and coulombs, show that 1 N/C = 1 V/m.

31. A uniform electric field has magnitude 240 N/C and is directed to the right. A particle with charge +4.2 nC moves along the straight line from

*a*to*b.*(a) What is the electric force that acts on the particle? (b) What is the work done on the particle by the electric field? (c) What is the potential difference*V*_{a}-*V*_{b}between points*a*and*b?*32. In a region where there is an electric field, the electric forces do +8.0 × 10

^{-19}J of work on an electron as it moves from point*X*to point*Y.*(a) Which point,*X*or*Y,*is at a higher potential? (b) What is the potential difference,*V*_{Y}-*V*_{X}, between point*Y*and point*X?***33.**Suppose a uniform electric field of magnitude 100.0 N/C exists in a region of space. How far apart are a pair of equipotential surfaces whose potentials differ by 1.0 V?

34. Draw some electric field lines and a few equipotential surfaces outside a negatively charged hollow conducting sphere. What shape are the equipotential surfaces?

**35.**Draw some electric field lines and a few equipotential surfaces outside a positively charged conducting cylinder. What shape are the equipotential surfaces?

36. It is believed that a large electric fish known as

*Torpedo occidentalis*uses electricity to shock its victims. A typical fish can deliver a potential difference of 0.20 kV for a duration of 1.5 ms. This pulse delivers charge at a rate of 18 C/s. (a) What is the rate at which work is done by the electric organs during a pulse? (b) What is the total amount of work done during one pulse?**37.**A positive point charge is located at the center of a hollow spherical metal shell with zero net charge. (a) Draw some electric field lines and sketch some equipotential surfaces for this arrangement. (b) Sketch graphs of the electric field magnitude and the potential as functions of*r.*

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** Problems 38 and 39.** A positively charged oil drop is injected into a region of uniform electric field between two oppositely charged, horizontally oriented plates spaced 16 cm apart.

38. If the electric force on the drop is found to be 9.6 × 10

^{-16}N and the potential difference between the plates is 480 V, what is the magnitude of the charge on the drop in terms of the elementary charge*e?*Ignore the small buoyant force on the drop.Tutorial: Millikan's Experiment

**39.**If the mass of the drop is 1.0 × 10^{-15}kg and it remains stationary when the potential difference between the plates is 9.76 kV, what is the magnitude of the charge on the drop? (Ignore the small buoyant force on the drop.)

17.4 Conservation of Energy for Moving Charges

40. Point

*P*is at a potential of 500.0 kV and point*S*is at a potential of 200.0 kV. The space between these points is evacuated. When a charge of +2*e*moves from*P*to*S,*by how much does its kinetic energy change?**41.**An electron is accelerated from rest through a potential difference Δ*V.*If the electron reaches a speed of 7.26 × 10^{6}m/s, what is the potential difference? Be sure to include the correct sign. (Does the electron move through an increase or a decrease in potential?)42. As an electron moves through a region of space, its speed decreases from 8.50 × 10

^{6}m/s to 2.50 × 10^{6}m/s. The electric force is the only force acting on the electron. (a) Did the electron move to a higher potential or a lower potential? (b) Across what potential difference did the electron travel?

**43.**In the electron gun of Example 17.8, if the potential difference between the cathode and anode is reduced to 6.0 kV, with what speed will the electrons reach the anode?44. In the electron gun of Example 17.8, if the electrons reach the anode with a speed of 3.0 × 10

^{7}m/s, what is the potential difference between the cathode and the anode?

**45.**A beam of electrons of mass*m*_{e}is deflected vertically by the uniform electric field between two oppositely charged, parallel metal plates. The plates are a distance*d*apart and the potential difference between the plates is Δ*V.*(a) What is the direction of the electric field between the plates? (b) If the*y*-component of the electrons' velocity as they leave the region between the plates is*v*_{y}, derive an expression for the time it takes each electron to travel through the region between the plates in terms of Δ*V,**v*_{y},*m*_{e},*d,*and*e.*(c) Does the electric potential energy of an electron increase, decrease, or stay constant while it moves between the plates? Explain.46. An electron (charge -

*e*) is projected horizontally into the space between two oppositely charged parallel plates. The electric field between the plates is 500.0 N/C upward. If the vertical deflection of the electron as it leaves the plates has magnitude 3.0 mm, how much has its kinetic energy increased due to the electric field? [*Hint:*First find the potential difference through which the electron moves.]

**47.**An alpha particle (charge +2*e*) moves through a potential difference Δ*V*= -0.50 kV. Its initial kinetic energy is 1.20 × 10^{-16}J. What is its final kinetic energy?48. In 1911, Ernest Rutherford discovered the nucleus of the atom by observing the scattering of helium nuclei from gold nuclei. If a helium nucleus with a mass of 6.68 × 10

^{-27}kg, a charge of +2*e,*and an initial velocity of 1.50 × 10^{7}m/s is projected head-on toward a gold nucleus with a charge of +79*e,*how close will the helium atom come to the gold nucleus before it stops and turns around? (Assume the gold nucleus is held in place by other gold atoms and does not move.)

**49.**The figure shows a graph of electric potential versus position along the*x*-axis. A proton is originally at point*A,*moving in the positive*x*-direction. How much kinetic energy does it need to have at point*A*in order to be able to reach point*E*(with no forces acting on the electron other than those due to the indicated potential)? Points*B,**C,*and*D*have to be passed on the way.50. Repeat Problem 49 for an electron rather than a proton.

17.5 Capacitors

**51.**A 2.0-μF capacitor is connected to a 9.0-V battery. What is the magnitude of the charge on each plate?52. The plates of a 15.0-μF capacitor have net charges of +0.75 μC and -0.75 μC, respectively. (a) What is the potential difference between the plates? (b) Which plate is at the higher potential?

**53.**If a capacitor has a capacitance of 10.2 μF and we wish to lower the potential difference across the plates by 60.0 V, what magnitude of charge will we have to remove from each plate?- Page 635
54. A parallel plate capacitor has a capacitance of 2.0 μF and plate separation of 1.0 mm. (a) How much potential difference can be placed across the capacitor before dielectric breakdown of air occurs (

*E*_{max}= 3 × 10^{6}V/m)? (b) What is the magnitude of the greatest charge the capacitor can store before breakdown?

**55.**A parallel plate capacitor is charged by connecting it to a 12-V battery. The battery is then disconnected from the capacitor. The plates are then pulled apart so the spacing between the plates is increased. What is the effect (a) on the electric field between the plates? (b) on the potential difference between the plates?56. A parallel plate capacitor has a capacitance of 1.20 nF. There is a charge of magnitude 0.800 μC on each plate. (a) What is the potential difference between the plates? (b) If the plate separation is doubled, while the charge is kept constant, what will happen to the potential difference?

**57.**A parallel plate capacitor is connected to a 12-V battery. While the battery remains connected, the plates are pushed together so the spacing is decreased. What is the effect on (a) the potential difference between the plates? (b) the electric field between the plates? (c) the magnitude of charge on the plates?Tutorial: Capacitor (parts a-b)

Tutorial: Capacitor (parts c-e)

Tutorial: Capacitor (parts f-g)

58. A parallel plate capacitor has a capacitance of 1.20 nF and is connected to a 12-V battery. (a) What is the magnitude of the charge on each plate? (b) If the plate separation is doubled while the plates remain connected to the battery, what happens to the charge on each plate and the electric field between the plates?

59. A variable capacitor is made of two parallel semicircular plates with air between them. One plate is fixed in place and the other can be rotated. The electric field is zero everywhere except in the region where the plates overlap. When the plates are directly across from one another, the capacitance is 0.694 pF. (a) What is the capacitance when the movable plate is rotated so that only one half its area is across from the stationary plate? (b) What is the capacitance when the movable plate is rotated so that two thirds of its area is across from the stationary plate?

60. A shark is able to detect the presence of electric fields as small as 1.0 μV/m. To get an idea of the magnitude of this field, suppose you have a parallel plate capacitor connected to a 1.5-V battery. How far apart must the parallel plates be to have an electric field of 1.0 μV/m between the plates?

**61.**Two metal spheres have charges of equal magnitude, 3.2 × 10^{-14}C, but opposite sign. If the potential difference between the two spheres is 4.0 mV, what is the capacitance? [*Hint:*The “plates” are not parallel, but the definition of capacitance holds.]62. Suppose you were to wrap the Moon in aluminum foil and place a charge

*Q*on it. What is the capacitance of the Moon in this case? [*Hint:*It is not necessary to have two oppositely charged conductors to have a capacitor. Use the definition of potential for a spherical conductor and the definition of capacitance to get your answer.]63. A tiny hole is made in the center of the negatively and positively charged plates of a capacitor, allowing a beam of electrons to pass through and emerge from the far side. If 40.0 V are applied across the capacitor plates and the electrons enter through the hole in the negatively charged plate with a speed of 2.50 × 10

^{6}m/s, what is the speed of the electrons as they emerge from the hole in the positive plate?64. A spherical conductor of radius

*R*carries a total charge*Q.*(a) Show that the magnitude of the electric field just outside the sphere is*E*= σ/ε_{0}, where σ is the charge per unit area on the conductor's surface. (b) Construct an argument to show why the electric field at a point*P*just outside*any*conductor in electrostatic equilibrium has magnitude*E*= σ/ε_{0}, where σ is the local surface charge density. [*Hint:*Consider a tiny area of an arbitrary conductor and compare it to an area of the same size on a spherical conductor with the same charge density. Think about the number of field lines starting or ending on the two areas.]

17.6 Dielectrics

**65.**A 6.2-cm by 2.2-cm parallel plate capacitor has the plates separated by a distance of 2.0 mm. (a) When 4.0 × 10^{-11}C of charge is placed on this capacitor, what is the electric field between the plates? (b) If a dielectric with dielectric constant of 5.5 is placed between the plates while the charge on the capacitor stays the same, what is the electric field in the dielectric?66. Before a lightning strike can occur, the breakdown limit for

*damp*air must be reached. If this occurs for an electric field of 3.33 × 10^{5}V/m, what is the maximum possible height above the Earth for the bottom of a thundercloud, which is at a potential 1.00 × 10^{8}V below Earth's surface potential, if there is to be a lightning strike?67. Two cows, with approximately 1.8 m between their front and hind legs, are standing under a tree during a thunderstorm. See the diagram with Conceptual Question 16. (a) If the equipotential surfaces about the tree just after a lightning strike are as shown, what is the average electric field between Cow A's front and hind legs? (b) Which cow is more likely to be killed? Explain.

68. A parallel plate capacitor has a charge of 0.020 μC on each plate with a potential difference of 240 V. The parallel plates are separated by 0.40 mm of bakelite. What is the capacitance of this capacitor?

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**69.**Two metal spheres are separated by a distance of 1.0 cm and a power supply maintains a constant potential difference of 900 V between them. The spheres are brought closer to one another until a spark flies between them. If the dielectric strength of dry air is 3.0 × 10^{6}V/m, what is the distance between the spheres at this time?

70. To make a parallel plate capacitor, you have available two flat plates of aluminum (area 120 cm

^{2}), a sheet of paper (thickness = 0.10 mm, κ = 3.5), a sheet of glass (thickness = 2.0 mm, κ = 7.0), and a slab of paraffin (thickness = 10.0 mm, κ = 2.0). (a) What is the largest capacitance possible using one of these dielectrics? (b) What is the smallest?**71.**A capacitor can be made from two sheets of aluminum foil separated by a sheet of waxed paper. If the sheets of aluminum are 0.30 m by 0.40 m and the waxed paper, of slightly larger dimensions, is of thickness 0.030 mm and dielectric constant κ = 2.5, what is the capacitance of this capacitor?

72. In capacitive electrostimulation, electrodes are placed on opposite sides of a limb. A potential difference is applied to the electrodes, which is believed to be beneficial in treating bone defects and breaks. If the capacitance is measured to be 0.59 pF, the electrodes are 4.0 cm

^{2}in area, and the limb is 3.0 cm in diameter, what is the (average) dielectric constant of the tissue in the limb?

17.7 Energy Stored in a Capacitor

**73.**A certain capacitor stores 450 J of energy when it holds 8.0 × 10^{-2}C of charge. What is (a) the capacitance of this capacitor and (b) the potential difference across the plates?74. What is the maximum electric energy density possible in dry air without dielectric breakdown occurring?

75. A parallel plate capacitor has a charge of 5.5 × 10

^{-7}C on one plate and -5.5 × 10^{-7}C on the other. The distance between the plates is increased by 50% while the charge on each plate stays the same. What happens to the energy stored in the capacitor?76. A large parallel plate capacitor has plate separation of 1.00 cm and plate area of 314 cm

^{2}. The capacitor is connected across a voltage of 20.0 V and has air between the plates. How much work is done on the capacitor as the plate separation is increased to 2.00 cm?**77.**Figure 17.31b shows a thundercloud before a lightning strike has occurred. The bottom of the thundercloud and the Earth's surface might be modeled as a charged parallel plate capacitor. The base of the cloud, which is roughly parallel to the Earth's surface, serves as the negative plate and the region of Earth's surface under the cloud serves as the positive plate. The separation between the cloud base and the Earth's surface is small compared to the length of the cloud. (a) Find the capacitance for a thundercloud of base dimensions 4.5 km by 2.5 km located 550 m above the Earth's surface. (b) Find the energy stored in this capacitor if the charge magnitude is 18 C.78. A parallel plate capacitor of capacitance 6.0 μF has the space between the plates filled with a slab of glass with κ = 3.0. The capacitor is charged by attaching it to a 1.5-V battery. After the capacitor is disconnected from the battery, the dielectric slab is removed. Find (a) the capacitance, (b) the potential difference, (c) the charge on the plates, and (d) the energy stored in the capacitor after the glass is removed.

**79.**A parallel plate capacitor is composed of two square plates, 10.0 cm on a side, separated by an air gap of 0.75 mm. (a) What is the charge on this capacitor when there is a potential difference of 150 V between the plates? (b) What energy is stored in this capacitor?80. The capacitor of Problem 79 is initially charged to a 150-V potential difference. The plates are then physically separated by another 0.750 mm in such a way that none of the charge can leak off the plates. Find (a) the new capacitance and (b) the new energy stored in the capacitor. Explain the result using conservation of energy.

**81.**Capacitors are used in many applications where you need to supply a short burst of energy. A 100.0-μF capacitor in an electronic flash lamp supplies an average power of 10.0 kW to the lamp for 2.0 ms. (a) To what potential difference must the capacitor initially be charged? (b) What is its initial charge?82. A parallel plate capacitor has a charge of 0.020 μC on each plate with a potential difference of 240 V. The parallel plates are separated by 0.40 mm of air. What energy is stored in this capacitor?

83. A parallel plate capacitor has a capacitance of 1.20 nF. There is a charge of 0.80 μC on each plate. How much work must be done by an external agent to double the plate separation while keeping the charge constant?

84. A defibrillator is used to restart a person's heart after it stops beating. Energy is delivered to the heart by discharging a capacitor through the body tissues near the heart. If the capacitance of the defibrillator is 9 μF and the energy delivered is to be 300 J, to what potential difference must the capacitor be charged?

**85.**A defibrillator consists of a 15-μF capacitor that is charged to 9.0 kV. (a) If the capacitor is discharged in 2.0 ms, how much charge passes through the body tissues? (b) What is the average power delivered to the tissues?

86. The bottom of a thundercloud is at a potential of -1.00 × 10

^{8}V with respect to Earth's surface. If a charge of -25.0 C is transferred to the Earth during a lightning strike, find the electric potential energy released. (Assume that the system acts like a capacitor—as charge flows, the potential difference decreases to zero.)- Page 637
87. (a) If the bottom of a thundercloud has a potential of -1.00 × 10

^{9}V with respect to Earth and a charge of -20.0 C is discharged from the cloud to Earth during a lightning strike, how much electric potential energy is released? (Assume that the system acts like a capacitor—as charge flows, the potential difference decreases to zero.) (b) If a tree is struck by the lightning bolt and 10.0% of the energy released vaporizes sap in the tree, about how much sap is vaporized? (Assume the sap to have the same latent heat as water.) (c) If 10.0% of the energy released from the lightning strike could be stored and used by a homeowner who uses 400.0 kW · hr of electricity per month, for how long could the lightning bolt supply electricity to the home?

Comprehensive Problems

88. Charges of -12.0 nC and -22.0 nC are separated by 0.700 m. What is the potential midway between the two charges?

**89.**Two point charges (+10.0 nC and -10.0 nC) are located 8.00 cm apart. (a) What is the electric potential energy when a point charge of -4.2 nC is placed at points*a,**b,*and*c*in turn? Let*U*= 0 when the -4.2 nC charge is far away (but the other two are still in place). (b) How much work would an external force have to do to move the point charge from*b*to*a?*90. If an electron moves from one point at a potential of -100.0 V to another point at a potential of +100.0 V, how much work is done by the electric field?

91. A van de Graaff generator has a metal sphere of radius 15 cm. To what potential can it be charged before the electric field at its surface exceeds 3.0 × 10

^{6}N/C (which is sufficient to break down dry air and initiate a spark)?92. Find the potential at the sodium ion, Na

^{+}, which is surrounded by two chloride ions, Cl^{-}, and a calcium ion, Ca^{2+}, in water as shown in the diagram. The effective charge of the positive sodium ion in water is 2.0 × 10^{-21}C, of the negative chlorine ion is -2.0 × 10^{-21}C, and of the positive calcium ion is 4.0 × 10^{-21}C.**93.**An infinitely long conducting cylinder sits near an infinite conducting sheet (side view in the diagram). The cylinder and sheet have equal and opposite charges; the cylinder is positive. (a) Sketch some electric field lines. (b) Sketch some equipotential surfaces.94. Two parallel plates are 4.0 cm apart. The bottom plate is charged positively and the top plate is charged negatively, producing a uniform electric field of 5.0 × 10

^{4}N/C in the region between the plates. What is the time required for an electron, which starts at rest at the upper plate, to reach the lower plate? (Assume a vacuum exists between the plates.)95. The potential difference across a cell membrane is −90 mV. If the membrane's thickness is 10 nm, what is the magnitude of the electric field in the membrane? Assume the field is uniform.

96. A beam of electrons traveling with a speed of 3.0 × 10

^{7}m/s enters a uniform, downward electric field of magnitude 2.0 × 10^{4}N/C between the deflection plates of an oscilloscope. The initial velocity of the electrons is perpendicular to the field. The plates are 6.0 cm long. (a) What is the direction and magnitude of the change in velocity of the electrons while they are between the plates? (b) How far are the electrons deflected in the ±*y*-direction while between the plates?**97.**A negatively charged particle of mass 5.00 × 10^{-19}kg is moving with a speed of 35.0 m/s when it enters the region between two parallel capacitor plates. The initial velocity of the charge is parallel to the plate surfaces and in the positive*x*-direction. The plates are square with a side of 1.00 cm and the voltage across the plates is 3.00 V. If the particle is initially 1.00 mm from both plates, and it just barely clears the positive plate after traveling 1.00 cm through the region between the plates, how many excess electrons are on the particle? You may ignore gravitational and edge effects.98. (a) Show that it was valid to ignore the gravitational force in Problem 97. (b) What are the components of velocity of the particle when it emerges from the plates?

99. Refer to Problem 97. One capacitor plate has an excess of electrons and the other has a matching deficit of electrons. What is the number of excess electrons?

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100. A parallel plate capacitor has a charge of 0.020 μC on each plate with a potential difference of 240 V. The parallel plates are separated by 0.40 mm of air. (a) What is the capacitance for this capacitor? (b) What is the area of a single plate? (c) At what voltage will the air between the plates become ionized? Assume a dielectric strength of 3.0 kV/mm for air.

**101.**A 200.0-μF capacitor is placed across a 12.0-V battery. When a switch is thrown, the battery is removed from the capacitor and the capacitor is connected across a heater that is immersed in 1.00 cm^{3}of water. Assuming that all the energy from the capacitor is delivered to the water, what is the temperature change of the water?102. A cell membrane has a surface area of 1.0 × 10

^{-7}m^{2}, a dielectric constant of 5.2, and a thickness of 7.5 nm. The membrane acts like the dielectric in a parallel plate capacitor; a layer of positive ions on the outer surface and a layer of negative ions on the inner surface act as the capacitor plates. The potential difference between the “plates” is 90.0 mV. (a) How much energy is stored in this capacitor? (b) How many positive ions are there on the outside of the membrane? Assume that all the ions are singly charged (charge +*e*).103. An axon has the outer part of its membrane positively charged and the inner part negatively charged. The membrane has a thickness of 4.4 nm and a dielectric constant κ = 5. If we model the axon as a parallel plate capacitor whose area is 5 μm

^{2}, what is its capacitance?104. An electron beam is deflected upward through 3.0 mm while traveling in a vacuum between two deflection plates 12.0 mm apart. The potential difference between the deflecting plates is 100.0 kV and the kinetic energy of each electron as it enters the space between the plates is 2.0 × 10

^{-15}J. What is the kinetic energy of each electron when it leaves the space between the plates?**105.**A point charge*q*= -2.5 nC is initially at rest adjacent to the negative plate of a capacitor. The charge per unit area on the plates is 4.0 μC/m^{2}and the space between the plates is 6.0 mm. (a) What is the potential difference between the plates? (b) What is the kinetic energy of the point charge just before it hits the positive plate, assuming no other forces act on it?106. An alpha particle (helium nucleus, charge +2

*e*) starts from rest and travels a distance of 1.0 cm under the influence of a uniform electric field of magnitude 10.0 kV/m. What is the final kinetic energy of the alpha particle?107. The inside of a cell membrane is at a potential of 90.0 mV lower than the outside. How much work does the electric field do when a sodium ion (Na

^{+}) with a charge of +*e*moves through the membrane from outside to inside?108. Draw some electric field lines and a few equipotential surfaces outside a positively charged metal cube. [

*Hint:*What shape are the equipotential surfaces close to the cube? What shape are they far away?]**109.**A parallel plate capacitor is attached to a battery that supplies a constant voltage. While the battery remains attached to the capacitor, the distance between the parallel plates increases by 25%. What happens to the energy stored in the capacitor?110. A parallel plate capacitor is attached to a battery that supplies a constant voltage. While the battery is still attached, a dielectric of dielectric constant κ = 3.0 is inserted so that it just fits between the plates. What is the energy stored in the capacitor after the dielectric is inserted in terms of the energy

*U*_{0}before the dielectric was inserted?111. (a) Calculate the capacitance per unit length of an axon of radius 5.0 μm (see Fig. 17.14). The membrane acts as an insulator between the conducting fluids inside and outside the neuron. The membrane is 6.0 nm thick and has a dielectric constant of 7.0. (

*Note:*The membrane is thin compared with the radius of the axon, so the axon can be treated as a parallel plate capacitor.) (b) In its resting state (no signal being transmitted), the potential of the fluid inside is about 85 mV lower than the outside. Therefore, there must be small net charges ±*Q*on either side of the membrane. Which side has positive charge? What is the magnitude of the charge density on the surfaces of the membrane?112. A 4.00-μF air gap capacitor is connected to a 100.0-V battery until the capacitor is fully charged. The battery is removed and then a dielectric of dielectric constant 6.0 is inserted between the plates without allowing any charge to leak off the plates. (a) Find the energy stored in the capacitor before and after the dielectric is inserted. [

*Hint:*First find the new capacitance and potential difference.] (b) Does an external agent have to do positive work to insert the dielectric or to remove the dielectric? Explain.**113.**It has only been fairly recently that 1.0-F capacitors have been readily available. A typical 1.0-F capacitor can withstand up to 5.00 V. To get an idea why it isn't easy to make a 1.0-F capacitor, imagine making a 1.0-F parallel plate capacitor using titanium dioxide (κ = 90.0, breakdown strength 4.00 kV/mm) as the dielectric. (a) Find the minimum thickness of the titanium dioxide such that the capacitor can withstand 5.00 V. (b) Find the area of the plates so that the capacitance is 1.0 F.114. The potential difference across a cell membrane from outside to inside is initially at −90 mV (when in its resting phase). When a stimulus is applied, Na

^{+}ions are allowed to move into the cell such that the potential changes to +20 mV for a short amount of time. (a) If the membrane capacitance per unit area is 1 μF/cm^{2}, how much charge moves through a membrane of area 0.05 cm^{2}? (b) The charge on Na^{+}is +*e.*How many ions move through the membrane?- Page 639
115. A parallel plate capacitor is connected to a battery. The space between the plates is filled with air. The electric field strength between the plates is 20.0 V/m. Then,

*with the battery still connected,*a slab of dielectric (κ = 4.0) is inserted between the plates. The thickness of the dielectric is half the distance between the plates. Find the electric field inside the dielectric.

Answers to Practice Problems

**17.1**(a) +0.018 J; (b) away from*Q;*(c)*U*decreases as the separation increases. The potential energy decrease accompanies an increase in kinetic energy as*q*moves faster and faster.**17.2**+0.064 J**17.3**the lower plate**17.4***V*_{B}= -1.5 × 10^{5}V; work (done by ) = -Δ*U*_{E}= -0.010 J**17.5**away from the +9.0-μC charge;*V*= 0**17.6**4 kV**17.7****17.8**-20.9 kV (Note that a positive charge gains kinetic energy when it moves through a potential decrease; a negative charge gains kinetic energy when it moves through a potential increase.)**17.9**8.9 nF; 18 μC; charge (capacitance is independent of potential difference)**17.10***C*doubles; maximum charge is unchanged**17.11**2.4 × 10^{5}ions**17.12**(a) 0.21 μC; (b) 85 μJ

Answers to Checkpoints

**17.1**Six pairs and therefore six terms in the potential energy (with subscripts 12, 13, 14, 23, 24, and 34).**17.2**points in the direction of*decreasing*potential, so the electric field is in the -*x*-direction.**17.3**The electric field magnitude is 25 V/m, so the potential decreases 25 V for each meter moved in the direction of the field. To move from one plane to another, the potential changes by 1.0 V and the distance must be**17.5**The magnitude of the charge on each plate is proportional to the potential difference between them. With one quarter the potential difference, the plates have one quarter as much charge: +0.12 C and -0.12 C. (The capacitance of the capacitor is*C*=*Q*/Δ*V*= 0.080 F.)