Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an online homework management program such as WileyPLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
Note: All charges are assumed to be point charges unless specified otherwise.
ssm Solution is in the Student Solutions Manual.
This icon represents a biomedical application.
Section 19.1 Potential Energy,
Section 19.2 The Electric Potential Difference

| 1 | ssm Suppose that the electric potential outside a living cell is higher than that inside the cell by 0.070 V. How much work is done by the electric force when a sodium ion (charge = +e) moves from the outside to the inside?

| 2 | During a particular thunderstorm, the electric potential difference between a cloud and the ground is , with the cloud being at the higher potential. What is the change in an electron’s electric potential energy when the electron moves from the ground to the cloud?

| 3 | Multiple-Concept Example 4 provides useful background for this problem. Point A is at a potential of +250 V, and point B is at a potential of −150 V. An -particle is a helium nucleus that contains two protons and two neutrons; the neutrons are electrically neutral. An α-particle starts from rest at A and accelerates toward B. When the -particle arrives at B, what kinetic energy (in electron volts) does it have?

| 4 | A particle has a charge of and moves from point A to point B, a distance of 0.20 m. The particle experiences a constant electric force, and its motion is along the line of action of the force. The difference between the particle’s electric potential energy at A and B is . (a) Find the magnitude and direction of the electric force that acts on the particle. (b) Find the magnitude and direction of the electric field that the particle experiences.

| 5 | Consult Interactive Solution 19.5 at to review one method for modeling this problem. Multiple-Concept Example 3 employs some of the concepts that are needed here. An electric car accelerates for 7.0 s by drawing energy from its 290-V battery pack. During this time, 1200 C of charge passes through the battery pack. Find the minimum horsepower rating of the car.

| 6 | Multiple-Concept Example 4 deals with the concepts that are important in this problem. As illustrated in Figure 19-6b, a negatively charged particle is released from rest at point B and accelerates until it reaches point A. The mass and charge of the particle are and , respectively. Only the gravitational force and the electrostatic force act on the particle, which moves on a horizontal straight line without rotating. The electric potential at A is 36 V greater than that at B; in other words, . What is the translational speed of the particle at point A?

| 7 | ssm Review Multiple-Concept Example 4 to see the concepts that are pertinent here. In a television picture tube, electrons strike the screen after being accelerated from rest through a potential difference of 25 000 V. The speeds of the electrons are quite large, and for accurate calculations of the speeds, the effects of special relativity must be taken into account. Ignoring such effects, find the electron speed just before the electron strikes the screen.

* | 8 | A typical 12-V car battery can deliver about of charge before dying. This is not very much. To get a feel for this, calculate the maximum number of kilograms of water (100 °C) that could be boiled into steam (100 °C) using energy from this battery.

* | 9 | ssm www The potential at location A is 452 V. A positively charged particle is released there from rest and arrives at location B with a speed vB. The potential at location C is 791 V, and when released from rest from this spot, the particle arrives at B with twice the speed it previously had, or 2vB. Find the potential at B.

** | 10 | A particle is uncharged and is thrown vertically upward from ground level with a speed of 25.0 m/s. As a result, it attains a maximum height h. The particle is then given a positive charge +q and reaches the same maximum height h when thrown vertically upward with a speed of 30.0 m/s. The electric potential at the height h exceeds the electric potential at ground level. Finally, the particle is given a negative charge −q. Ignoring air resistance, determine the speed with which the negatively charged particle must be thrown vertically upward, so that it attains exactly the maximum height h. In all three situations, be sure to include the effect of gravity.
Section 19.3 The Electric Potential Difference Created by Point Charges

| 11 | ssm Two charges A and B are fixed in place, at different distances from a certain spot. At this spot the potentials due to the two charges are equal. Charge A is 0.18 m from the spot, while charge B is 0.43 m from it. Find the ratio qB/qA of the charges.

| 12 | Point A is located 0.25 m away from a charge of . Point B is located 0.50 m away from the charge. What is the electric potential difference between these two points?

| 13 | An electron and a proton are initially very far apart (effectively an infinite distance apart). They are then brought together to form a hydrogen atom, in which the electron orbits the proton at an average distance of . What is , which is the change in the electric potential energy?

| 14 | Location A is 3.00 m to the right of a point charge q. Location B lies on the same line and is 4.00 m to the right of the charge. The potential difference between the two locations is . What is the magnitude and sign of the charge?

| 15 | ssm www Two identical point charges are fixed to diagonally opposite corners of a square that is 0.500 m on a side. Each charge is . How much work is done by the electric force as one of the charges moves to an empty corner?

| 16 | The drawing shows six point charges arranged in a rectangle. The value of q is , and the distance d is 0.13 m. Find the total electric potential at location P, which is at the center of the rectangle.

| 17 | Review Conceptual Example 7 as background for this problem. One charges are fixed in place with a separation d. One charge is positive and has twice the magnitude of the other charge, which is negative. The positive charge lies to the left of the negative charge, as in Figure 19-10. Relative to the negative charge, locate the two spots on the line through the charges where the total potential is zero.

| 18 | Identical charges are fixed to adjacent corners of a square. What charge (magnitude and algebraic sign) should be fixed to one of the empty corners, so that the total electric potential at the remaining empty corner is 0 V?

* | 19 | ssm A charge of is fixed in place. From a horizontal distance of 0.0450 m, a particle of mass and charge is fired with an initial speed of 65.0 m/s directly toward the fixed charge. How far does the particle travel before its speed is zero?

* | 20 | Four identical charges ( each) are brought from infinity and fixed to a straight line. The charges are located 0.40 m apart. Determine the electric potential energy of this group.

* | 21 | Refer to Interactive Solution 19.21 at to review one way in which this problem can be solved. Two protons are moving directly toward one another. When they are very far apart, their initial speeds are . What is the distance of closest approach?

* | 22 | Identical point charges of are fixed to diagonally opposite corners of a square. A third charge is then fixed at the center of the square, such that it causes the potentials at the empty corners to change signs without changing magnitudes. Find the sign and magnitude of the third charge.

* | 23 | ssm Determine the electric potential energy for the array of three charges shown in the drawing, relative to its value when the charges are infinitely far away.

Problem 23

** | 24 | A positive charge +q1 is located to the left of a negative charge −q2. On a line passing through the two charges, there are two places where the total potential is zero. The first place is between the charges and is 4.00 cm to the left of the negative charge. The second place is 7.00 cm to the right of the negative charge. (a) What is the distance between the charges? (b) Find q1/q2, the ratio of the magnitudes of the charges.

** | 25 | Charges q1 and q2 are fixed in place, q2 being located at a distance d to the right of q1. A third charge q3 is then fixed to the line joining q1 and q2 at a distance d to the right of q2. The third charge is chosen so the potential energy of the group is zero; that is, the potential energy has the same value as that of the three charges when they are widely separated. Determine the value for q3, assuming that (a) q1 = q2 = q and (b) q1 = q and q2 = −q. Express your answers in terms of q.

** | 26 | One particle has a mass of and a charge of . A second particle has a mass of and the same charge. The two particles are initially held in place and then released. The particles fly apart, and when the separation between them is 0.100 m, the speed of the particle is 125 m/s. Find the initial separation between the particles.
Section 19.4 Equipotential Surfaces and Their Relation to the Electric Field

| 27 | ssm A spark plug in an automobile engine consists of two metal conductors that are separated by a distance of 0.75 mm. When an electric spark jumps between them, the magnitude of the electric field is . What is the magnitude of the potential difference between the conductors?

| 28 | An equipotential surface that surrounds a point charge q has a potential of 490 V and an area of 1.1 m2. Determine q.

| 29 | Two equipotential surfaces surround a point charge. How far is the 190-V surface from the 75.0-V surface?

| 30 | The inner and outer surfaces of a cell membrane carry a negative and positive charge, respectively. Because of these charges, a potential difference of about 0.070 V exists across the membrane. The thickness of the cell membrane is . What is the magnitude of the electric field in the membrane?

| 31 | ssm www When you walk across a rug on a dry day, your body can become electrified, and its electric potential can change. When the potential becomes large enough, a spark of negative charges can jump between your hand and a metal surface. A spark occurs when the electric field strength created by the charges on your body reaches the dielectric strength of the air. The dielectric strength of the air is and is the electric field strength at which the air suffers electrical breakdown. Suppose a spark 3.0 mm long jumps between your hand and a metal doorknob. Assuming that the electric field is uniform, find the potential difference between your hand and the doorknob.

* | 32 | The drawing shows the potential at five points on a set of axes. Each of the four outer points is from the point at the origin. From the data shown, find the magnitude and direction of the electric field in the vicinity of the origin.

| 33 | Refer to Interactive Solution 19.33 at to review a method by which this problem can be solved. The electric field has a constant value of and is directed downward. The field is the same everywhere. The potential at a point P within this region is 155 V. Find the potential at the following points: (a) directly above P, (b) directly below P, (c) directly to the right of P.

* | 34 | The drawing shows the electric potential as a function of distance along the x axis. Determine the magnitude of the electric field in the region (a) A to B, (b) B to C, and (c) C to D.

* | 35 | ssm Equipotential surface A has a potential of 5650 V, while equipotential surface B has a potential of 7850 V. A particle has a mass of and a charge of . The particle has a speed of 2.00 m/s on surface A. An outside force is applied to the particle, and it moves to surface B, arriving there with a speed of 3.00 m/s. How much work is done by the outside force in moving the particle from A to B?
Section 19.5 Capacitors and Dielectrics

| 36 | An axon is the relatively long tail-like part of a neuron, or nerve cell. The outer surface of the axon membrane (dielectric constant = 5, ) is charged positively, and the inner portion is charged negatively. Thus, the membrane is a kind of capacitor. Assuming that an axon can be treated like a parallel plate capacitor with a plate area of , what is its capacitance?

| 37 | ssm The membrane that surrounds a certain type of living cell has a surface area of and a thickness of . Assume that the membrane behaves like a parallel plate capacitor and has a dielectric constant of 5.0. (a) The potential on the outer surface of the membrane is +60.0 mV greater than that on the inside surface. How much charge resides on the outer surface? (b) If the charge in part (a) is due to K+ ions (charge +e), how many such ions are present on the outer surface?

| 38 | What voltage is required to store of charge on the plates of a capacitor?

| 39 | Refer to Interactive Solution 19.39 at for one approach to this problem. The electronic flash attachment for a camera contains a capacitor for storing the energy used to produce the flash. In one such unit, the potential difference between the plates of an capacitor is 280 V. (a) Determine the energy that is used to produce the flash in this unit. (b) Assuming that the flash lasts for , find the effective power or “wattage” of the flash.

| 40 | A capacitor stores of charge when connected to a 6.0-V battery. How much charge does the capacitor store when connected to a 9.0-V battery?

| 41 | ssm A parallel plate capacitor has a capacitance of when filled with a dielectric. The area of each plate is 1.5 m2 and the separation between the plates is . What is the dielectric constant of the dielectric?

| 42 | Two capacitors are identical, except that one is empty and the other is filled with a dielectric . The empty capacitor is connected to a 12.0-V battery. What must be the potential difference across the plates of the capacitor filled with a dielectric such that it stores the same amount of electrical energy as the empty capacitor?

* | 43 | ssm Interactive LearningWare 19.2 at reviews the concepts pertinent to this problem. What is the potential difference between the plates of a 3.3-F capacitor that stores sufficient energy to operate a 75-W light bulb for one minute?

* | 44 | An empty parallel plate capacitor is connected between the terminals of a 9.0-V battery and charged up. The capacitor is then disconnected from the battery, and the spacing between the capacitor plates is doubled. As a result of this change, what is the new voltage between the plates of the capacitor?

* | 45 | Refer to Interactive Solution 19.45 at and Multiple-Concept Example 10 for help in solving this problem. An empty capacitor has a capacitance of and is connected to a 12-V battery. A dielectric material is inserted between the plates of this capacitor. What is the magnitude of the surface charge on the dielectric that is adjacent to either plate of the capacitor? (Hint: The surface charge is equal to the difference in the charge on the plates with and without the dielectric.)

* | 46 | Review Conceptual Example 11 before attempting this problem. An empty capacitor is connected to a 12.0-V battery and charged up. The capacitor is then disconnected from the battery, and a slab of dielectric material is inserted between the plates. Find the amount by which the potential difference across the plates changes. Specify whether the change is an increase or a decrease.

** | 47 | ssm The drawing shows a parallel plate capacitor. One-half of the region between the plates is filled with a material that has a dielectric constant . The other half is filled with a material that has a dielectric constant . The area of each plate is A, and the plate separation is d. The potential difference across the plates is V. Note especially that the charge stored by the capacitor is , where q1 and q2 are the charges on the area of the plates in contact with materials 1 and 2, respectively. Show that .

** | 48 | The plate separation of a charged capacitor is 0.0800 m. A proton and an electron are released from rest at the midpoint between the plates. Ignore the attraction between the two particles, and determine how far the proton has traveled by the time the electron strikes the positive plate.

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