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© Copyright 2014 Pearson Education, Inc. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
17-1
Responses to Questions
1. If two points are at the same potential, then no NET work is done in moving a test charge from one point to the other. Along some segments of the path, some positive work might be done, but along other segments of the path, negative work would be done. And if the object is moved strictly along an equipotential line, then no work would be done along any segment of the path. Along any segment of the path where positive or negative work is done, a force would have to be exerted. If the object is moved along an equipotential line, then no force would be exerted along that segment of the path. This is analogous to climbing up and then back down a flight of stairs to get from one point to another point on the same floor of a building. Gravitational potential increased while going up the stairs, and decreased while going down the stairs. A force was required both to go up the stairs and to go down the stairs. If instead you walked on the level from one point to another, then the gravitational potential was constant, and no force was needed to change gravitational potential. 2. A negative charge will move toward a region of higher potential. A positive charge will move toward a region of lower potential. The potential energy of both charges decreases as they move, because they gain kinetic energy. 3. (
a
) Electric potential, a scalar, is the electric potential energy per unit charge at a point in space. Electric field, a vector, is the electric force per unit charge at a point in space. (
b
) Electric potential energy is the work done against the electric force in moving a charge from a specified location of zero potential energy to some other location. Electric potential is the electric potential energy per unit charge. 4. The potential energy of the electron is proportional to the voltage used to accelerate it. Thus, if the voltage is multiplied by a factor of 4, then the potential energy is increased by a factor of 4 also. Then, by energy conservation, we assume that all of the potential energy is converted to kinetic energy during the acceleration process. Thus the kinetic energy has increased by a factor of 4 also. Finally, since the speed is proportional to the square root of kinetic energy, the speed must increase by a factor of 2.
E
LECTRIC
P
OTENTIAL
17
17-2 Chapter 17
© Copyright 2014 Pearson Education, Inc. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
5. The electric field is zero at the midpoint of the line segment joining the two equal positive charges. The electric field due to each charge is of the same magnitude at that location, because the location is equidistant from both charges, but the two fields are in the opposite direction. Thus the net electric field is zero there. The electric potential is never zero along that line, except at infinity. The electric potential due to each charge is positive, so the total potential, which is the algebraic sum of the two potentials, is always positive. 6. A negative particle will have its electric potential energy decrease if it moves from a region of low electric potential to one of high potential. By Eq. 17–3, if the charge is negative and the potential difference is positive, then the change in potential energy will be negative and therefore decrease. 7. There is no general relationship between the value of
V
and the value of .
E
Instead, the magnitude of
E
is equal to the rate at which
V
decreases over a short distance. Consider the point midway between two positive charges.
E
is 0 there, but
V
is high. Or consider the point midway between two negative charges.
E
is also 0 there, but
V
is low, because it is negative. Finally, consider the point midway between positive and negative charges of equal magnitude. There
E
is not 0, because it points toward the negative charge, but
V
is zero. 8. Two equipotential lines cannot cross. That would indicate that a region in space had two different values for the potential. For example, if a 40-V line and a 50-V line crossed, then the potential at the point of crossing would be both 40 V and 50 V, which is impossible. As an analogy, imagine contour lines on a topographic map. They also never cross because one point on the surface of the Earth cannot have two different values for elevation above sea level. Likewise, the electric field is perpendicular to the equipotential lines. If two lines crossed, then the electric field at that point would point in two different directions simultaneously, which is not possible. 9. The equipotential lines (in black) are perpendicular to the electric field lines (in red). 10. Any imbalance of charge that exists would quickly be resolved. Suppose the positive plate, connected to the positive terminal of the battery, had more charge than the negative plate. Then negative charges from the negative battery terminal would be attracted to the negative plate by the more charged positive plate. This would continue only until the negative plate had the same magnitude of charge as the positive plate. If the negative plate became “overcharged,” then the opposite transfer of charge would take place, again until equilibrium was reached. Another way to explain the balance of charge is that neither the battery nor the capacitor can create or destroy charge. Since they were neutral before they were connected, they must be neutral after they are connected. The charge removed from one plate appears as excess charge on the other plate. This is true regardless of the conductor size or shape. 11. (
a
) Once the two spheres are placed in contact with each other, they effectively become one larger conductor. They will have the same potential because the potential everywhere on a conducting surface is constant. (
b
) Because the spheres are identical in size, an amount of charge /2
Q
will flow from the initially charged sphere to the initially neutral sphere so that they will have equal charges.
Electric Potential 17-3
© Copyright 2014 Pearson Education, Inc. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
12. A force is required to increase the separation of the plates of an isolated capacitor because you are pulling a positive plate away from a negative plate. Since unlike charges attract, it takes a force to move the oppositely charged plates apart. The work done in increasing the separation goes into increasing the electric potential energy stored between the plates. The capacitance decreases, and the potential between the plates increases since the charge has to remain the same. 13. If the electric field in a region of space is uniform, then you can infer that the electric potential is increasing or decreasing uniformly in that region. For example, if the electric field is 10 V/m in a region of space, then you can infer that the potential difference between two points 1 meter apart (measured parallel to the direction of the field) is 10 V. If the electric potential in a region of space is uniform, then you can infer that the electric field there is zero. 14. The electric potential energy of two unlike charges is negative if we take the 0 location for potential energy to be when the charges are infinitely far apart. The electric potential energy of two like charges is positive. In the case of unlike charges, work must be done to separate the charges. In the case of like charges, work must be done to move the charges together. 15. (
c
) If the voltage across a capacitor is doubled, then the amount of energy it can store is quadrupled:
21PE2
.
CV
=
16. (
a
) If the capacitor is isolated, then
Q
remains constant, and
21PE2
QC
=
becomes
21PE2
Q KC
′
=
and the stored energy decreases. (
b
) If the capacitor remains connected to a battery so
V
does not change, then
21PE2
CV
=
becomes
21PE2
KCV
′
=
and the stored energy increases. 17. (
a
) When the dielectric is removed, the capacitance decreases by a factor of
K
. (
b
) The charge decreases since
Q CV
=
and the capacitance decreases while the potential difference remains constant. The “lost” charge returns to the battery. (
c
) The potential difference stays the same because it is equal to the battery voltage. (
d
) If the potential difference remains the same and the capacitance decreases, then the energy stored in the capacitor must also decrease, since
21PE2
.
CV
=
(
e
) The electric field between the plates will stay the same because the potential difference across the plates and the distance between the plates remain constant. 18. We meant that the capacitance did not depend on the amount of charge stored or on the potential difference between the capacitor plates. Changing the amount of charge stored or the potential difference will not change the capacitance.
Responses to MisConceptual Questions
1. (
b
) The two different concepts, electric potential and electric potential energy, are often confused, since their names are so similar. The electric potential is determined by the electric field and is independent of the charge placed in the field. Therefore, doubling the charge will not affect the electric potential. The electric potential energy is the product of the electric potential and the electric charge. Doubling the charge will double the electric potential energy.
17-4 Chapter 17
© Copyright 2014 Pearson Education, Inc. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
2. (
a
) It is very important to realize that the electric field is a vector, while the electric potential is a scalar. The electric field at the point halfway between the two charges is the sum of the electric fields from each charge. The magnitudes of the fields will be the same, but they will point in opposite directions. Their sum is therefore zero. The electric potential from each charge is positive. Since the potential is a scalar, the net potential at the midpoint is the sum of the two potentials, which will also be positive. 3. (
b
) The net electric field at the center is the vector sum of the electric fields due to each charge. The fields will have equal magnitudes at the center, but the fields from the charges at opposite corners point in opposite directions, so the net field will be zero. The electric potential from each charge is a nonzero scalar. At the center the magnitudes of the four potentials are equal and sum to a nonzero value. 4. (
d
) A common misconception is that the electric field and electric potential are proportional to each other. However, the electric field is proportional to the change in the electric potential. If the electric potential is constant (no change), then the electric field must be zero. The electric potential midway between equal but opposite charges is zero, but the electric field is not zero at that point, so (
a
) is incorrect. The electric field midway between two equal positive charges is zero, but the electric potential is not zero at that point, so (
b
) is incorrect. The electric field between two oppositely charged parallel plates is constant, but the electric potential decreases as one moves from the positive plate toward the negative plate, so (
c
) is incorrect. 5. (
b
) It is commonly thought that it only takes 2
W
to bring together the three charges. However, it takes
W
to bring two charges together. When the third charge is brought in, it is repelled by both of the other charges. It therefore takes an additional 2
W
to bring in the third charge. Adding the initial work to bring the first two charges together gives a total work of 3
W
. 6. (
e
) A common misconception is that the electron feels the greater force because it experiences the greater acceleration. However, the magnitude of the force is the product of the electric field and the charge. Since both objects have the same magnitude of charge and are in the same electric field, they will experience the same magnitude force (but in opposite directions). Since the electron is lighter, it will experience a greater acceleration. 7. (
c
) It is often thought that the electron, with its greater acceleration and greater final speed, will have the greater final kinetic energy. However, the increase in kinetic energy is equal to the decrease in potential energy, which is the product of the object’s charge and the change in potential. However, in this situation, the increase in kinetic energy is equal to the decrease in electric potential energy, which is the product of the object’s charge and the change in electric potential through which it passes. The change in potential for each object has the same magnitude, but they have opposite signs since they move in opposite directions in the field. The magnitude of the charge of each object is the same, but they have opposite signs; thus they experience the same change in electric potential energy and therefore they have the same final kinetic energy. 8. (
d
), (
e
) The capacitance is determined by the shape of the capacitor (area of plates and separation distance) and the material between the plates (dielectric). The capacitance is the constant ratio of the charge on the plates to the potential difference between them. Changing the charge will change the potential difference but not the capacitance, so (
d
) is one correct answer. Changing the energy stored in a capacitor will change the charge on the plates and the potential difference between them, but will not change the capacitance, so (
e
) is also correct.

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