19.3

The Electric Potential Difference Created by Point Charges

A positive point charge +q creates an electric potential in a way that Figure 19-7 helps explain. This picture shows two locations A and B, at distances rA and rB from the charge. At any position between A and B an electrostatic force of repulsion acts on a positive test charge +q0. The magnitude of the force is given by Coulomb’s law as , where we assume for convenience that q0 and q are positive, so that and . When the test charge moves from A to B, work is done by this force. Since r varies between rA and rB, the force F also varies, and the work is not the product of the force and the distance between the points. (Recall from Section 6.1 that work is force times distance only if the force is constant.) However, the work WAB can be found with the methods of integral calculus. The result is

This result is valid whether q is positive or negative, and whether q0 is positive or negative. The potential difference, , between A and B can now be determined by substituting this expression for WAB into Equation 19.4:

| (19.5)

As point B is located farther and farther from the charge q, rB becomes larger and larger. In the limit that rB is infinitely large, the term kq/rB becomes zero, and it is customary to set VB equal to zero also. In this limit, Equation 19.5 becomes , and it is standard convention to omit the subscripts and write the potential in the following form:

| (19.6)

The symbol V in this equation does not refer to the potential in any absolute sense. Rather, stands for the amount by which the potential at a distance r from a point charge differs from the potential at an infinite distance away. In other words, V refers to a potential difference with the arbitrary assumption that the potential at infinity is zero.

Figure 19-7 The positive test charge +q0 experiences a repulsive force due to the positive point charge +q. As a result, work is done by this force when the test charge moves from A to B. Consequently, the electric potential is higher (uphill) at A and lower (downhill) at B.

With the aid of Equation 19.6, we can describe the effect that a point charge q has on the surrounding space. When q is positive, the value of is also positive, indicating that the positive charge has everywhere raised the potential above the zero reference value. Conversely, when q is negative, the potential V is also negative, indicating that the negative charge has everywhere decreased the potential below the zero reference value. The next example deals with these effects quantitatively.

| Example 5 | | | The Potential of a Point Charge

Using a zero reference potential at infinity, determine the amount by which a point charge of alters the electric potential at a spot 1.2 m away when the charge is (a) positive and (b) negative. Reasoning A point charge q alters the potential at every location in the surrounding space. In the expression , the effect of the charge in increasing or decreasing the potential is conveyed by the algebraic sign for the value of q.
Solution

a. | Figure 19-8a shows the potential when the charge is positive:

| (19.6)

b. | Part b of the drawing illustrates the results when the charge is negative. A calculation similar to the one in part (a) shows that the potential is now negative: .

Figure 19-8 A point charge of 4.0 × 10−8 C alters the potential at a spot 1.2 m away. The potential is (a) increased by 300 V when the charge is positive and (b) decreased by 300 V when the charge is negative, relative to a zero reference potential at infinity.

A single point charge raises or lowers the potential at a given location, depending on whether the charge is positive or negative. When two or more charges are present, the potential due to all the charges is obtained by adding together the individual potentials, as the next two examples show.

Problem solving insight

| Example 6 | | | The Total Electric Potential

At locations A and B in Figure 19-9, find the total electric potential due to the two point charges.

Figure 19-9 Both the positive and negative charges affect the electric potential at locations A and B.

Reasoning At each location, each charge contributes to the total electric potential. We obtain the individual contributions by using and find the total potential by adding the individual contributions algebraically. The two charges have the same magnitude, but different signs. Thus, at A the total potential is positive because this spot is closer to the positive charge, whose effect dominates over that of the more distant negative charge. At B, midway between the charges, the total potential is zero, since the potential of one charge exactly offsets that of the other.
Solution

Location | Contribution from + Charge |  | Contribution from – Charge | Total Potential
A |  | + |  |
B |  | + |  |

| Conceptual Example 7 | | | Where is the Potential Zero?

Two point charges are fixed in place, as in Figure 19-10. The positive charge is +2q and has twice the magnitude of the negative charge, which is −q. On the line that passes through the charges, how many places are there at which the total potential is zero?

Figure 19-10 Two point charges, one positive and one negative. The positive charge, +2q, has twice the magnitude of the negative charge, −q.

Reasoning and Solution The total potential is the algebraic sum of the individual potentials created by each charge. It will be zero if the potential due to the positive charge is exactly offset by the potential due to the negative charge. The potential of a point charge is directly proportional to the charge and inversely proportional to the distance from the charge. The positive charge has the larger magnitude and, in the region to its left, is closer to any spot than the negative charge is. As a result, the potential of the positive charge here always dominates over that of the negative charge. Therefore, the total potential cannot be zero anywhere to the left of the positive charge.
Between the charges there is a location where the individual potentials do balance. We saw a similar situation in Example 6, where the balance occurred at the midpoint between two charges that had equal magnitudes. Now the charges have unequal magnitudes, so the balance does not occur at the midpoint. Instead, it occurs at a location that is closer to the charge with the smaller magnitude—namely, the negative charge. Then, since the potential of a point charge is inversely proportional to the distance from the charge, the effect of the smaller charge will be able to offset that of the more distant larger charge.

Problem solving insight At any location, the total electric potential is the algebraic sum of the individual potentials created by each point charge that is present.

To the right of the negative charge, there is another location at which the individual potentials exactly offset one another. All places on this section of the line are closer to the negative than to the positive charge. In this region, therefore, a location exists at which the potential of the smaller negative charge again exactly offsets that of the more distant and larger positive charge.
Related Homework: Problem 17

In Example 6 we determined the total potential at a spot due to several point charges. In Example 8 we now extend this technique to find the total potential energy of three charges.

| Example 8 | | | The Potential Energy of a Group of Charges

Figure 19-11 shows three point charges; initially, they are infinitely far apart. They are then brought together and placed at the corners of an equilateral triangle. Each side of the triangle has a length of 0.50 m. Determine the electric potential energy of the triangular group. In other words, determine the amount by which the electric potential energy of the group differs from that of the three charges in their initial, infinitely separated locations.

Figure 19-11 Three point charges are placed on the corners of an equilateral triangle. Example 8 illustrates how to determine the total electric potential energy of this group of charges.

Reasoning We will proceed in steps by adding charges to the triangle, one at a time, and then determining the electric potential energy at each step. According to Equation 19.3, , the electric potential energy is the product of the charge and the electric potential at the spot where the charge is placed. The total electric potential energy of the triangular group is the sum of the energies of each step in assembling the group.
Solution The order in which the charges are put on the triangle does not matter; we begin with the charge of . When this charge is placed at a corner of the triangle, it has no electric potential energy, according to . This is because the total potential V produced by the other two charges is zero at this corner, since they are infinitely far away. Once the charge is in place, the potential it creates at either empty corner is

| (19.6)

Therefore, when the charge is placed at the second corner of the triangle, its electric potential energy is

| (19.3)

The electric potential at the remaining empty corner is the sum of the potentials due to the two charges that are already in place:

When the third charge, , is placed at the remaining empty corner, its electric potential energy is

| (19.3)

The total potential energy of the triangular group differs from that of the widely separated charges by an amount that is the sum of the potential energies calculated above:

This energy originates in the work done to bring the charges together.

Problem solving insight Be careful to distinguish between the concepts of potential V and electric potential energy EPE. Potential is electric potential energy per unit charge: .

Check Your Understanding 2

The figure shows four arrangements of two point charges. In each arrangement consider the total electric potential that the charges produce at location P. Rank the values of the total potential in descending order (largest first). (The answer is given at the end of the book.)

Background: The electric potential produced by a point charge is the key concept here. The potential may be positive or negative, and its magnitude depends on the magnitude of the charge as well as the distance between it and the location P.
For similar questions (including calculational counterparts), consult Self-Assessment Test 19.1, which is described next.

Test your understanding of the material in Sections 19.1, 19.2 and 19.3:

· | Electric Potential Energy
· | Electric Potential Difference
· | Electric Potential Difference Created by Point Charges
· | Electric Potential Energy of a Group of Point Charges

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.