Electric Potential and Potential Energy due to Point Charges
Consider an isolated positive point charge q. Recall that such a charge produces an electric field that is directed radially outward from the charge. To find the electric potential at a point located a distance r from the charge, we begin with the general expression for potential difference:
where A and B are the two arbitrary points shown in Figure 25.6. At any field point, the electric field due to the point charge is E = keqr̂/r2 (Eq. 23.4), where r̂ is a unit vector directed from the charge toward the field point. The quantity E.ds can be expressed as
Because the magnitude of r̂ is 1, the dot product r̂.ds = ds cosθ, where θ is the angle between r̂. and ds. Furthermore, ds cos θ is the projection of ds onto r ; thus, ds cos θ = dr. That is, any displacement ds along the path from point A to point B produces a change dr in the magnitude of r, the radial distance to the charge creating the field. Making these substitutions, we find that E.ds = ( keqr̂/r2 )dr; hence, the expression for the potential difference becomes
The integral of E.ds is independent of the path between points A and B - as it must be because the electric field of a point charge is conservative. Furthermore, Equation 25.10 expresses the important result that the potential difference between any two points A and B in a field created by a point charge depends only on the radial coordinates rA and rB . It is customary to choose the reference of electric potential to be zero at rA = ∞. With this reference, the electric potential created by a point charge at any distance r from the charge is
Electric potential is graphed in Figure 25.7 as a function of r, the radial distance from a positive charge in the xy plane. Consider the following analogy to gravitational potential: Imagine trying to roll a marble toward the top of a hill shaped like Figure 25.7a. The gravitational force experienced by the marble is analogous to the repulsive force experienced by a positively charged object as it approaches another positively charged object. Similarly, the electric potential graph of the region surrounding a negative charge is analogous to a “hole” with respect to any approaching positively charged objects. A charged object must be infinitely distant from another charge before the surface is “flat” and has an electric potential of zero.
We obtain the electric potential resulting from two or more point charges by applying the superposition principle. That is, the total electric potential at some point P due to several point charges is the sum of the potentials due to the individual charges. For a group of point charges, we can write the total electric potential at P in the form
where the potential is again taken to be zero at infinity and ri is the distance from the point P to the charge qi . Note that the sum in Equation 25.12 is an algebraic sum of scalars rather than a vector sum (which we use to calculate the electric field of a group of charges). Thus, it is often much easier to evaluate V than to evaluate E. The electric potential around a dipole is illustrated in Figure 25.8.
We now consider the potential energy of a system of two charged particles. If V1 is the electric potential at a point P due to charge q1 , then the work an external agent must do to bring a second charge q2 from infinity to P without acceleration is q2V1. By definition, this work equals the potential energy U of the two-particle system when the particles are separated by a distance r12 (Fig. 25.9). Therefore, we can express the potential energy as
Note that if the charges are of the same sign, U is positive. This is consistent with the fact that positive work must be done by an external agent on the system to bring the two charges near one another (because like charges repel). If the charges are of opposite sign, U is negative; this means that negative work must be done against the attractive force between the unlike charges for them to be brought near each other.
If more than two charged particles are in the system, we can obtain the total potential energy by calculating U for every pair of charges and summing the terms algebraically. As an example, the total potential energy of the system of three charges shown in Figure 25.10 is
Physically, we can interpret this as follows: Imagine that q1 is fixed at the position shown in Figure 25.10 but that q2 and q3 are at infinity. The work an external agent must do to bring q2 from infinity to its position near q1 is keq1q2/ r12 which is the first term in Equation 25.14. The last two terms represent the work required to bring q3 from infinity to its position near q1 and q2 . (The result is independent of the order in which the charges are transported.)