Potential Energy of Point Charges

Consider a point charge q placed at position where the potential is V. The potential energy associated with the interaction of this single charge with the charges that created V is

U = qV (1.19)

If the source of the potential is a point charge Q, the potential at a distance r from Q is
V = kQ/r. Therefore, the potential energy shared by two charges q and Q separated by r is

U =  kqQ/r (1.20)

Implicit in Equation (1.20) is the choice U = 0 at r = ∞, which allows the following interpretation.

The potential energy of the system of two charges is the external work needed to bring the charges from infinity to the separation r without a change in kinetic energy.

When both charges have the same sign, their potential energy is positive: Positive work is needed to reduce their separation against their mutual repulsion. When the charges have opposite signs, the external work is negative. In this case, the external force has to prevent the particles from speeding up−which means that the external force is directed opposite to the displacement. Negative potential energy means that external work is required to separate the charges.

When calculating the total potential energy of a system of several charges, it is better to write equation (1.20) as

Uij = kq_iq_j/r_ij  (1.21)

This form helps us not to double count the contributions of the charges.

Note that U_ij = U_ji and that we do not include terms for which . Since the potentials obey the principle of linear superposition the total potential energy of a system is simply an algebraic sum and does not depend on how the charges are assembled.

Solution

(a) The total potential at the point P is the scalar sum

VP = V1 + V2 + V3 = 

Using the given values

V1 = 

Similarly, V2 = −3.6 × 103 V and V3 = 9 × 103 V

The total potential is VP = 7.65 × 103 V

(b) The External work is Wext = q (Vf − Vi). In this case Vi = 0, so

Wext = q4VP = (2.5 × 10−6C) (7.65 × 103 V) = 0.019 J

(c) The total potential energy of the three charges in the (scalar) sum

U = U12 + U13 + U23 = 

We find, for example

U12 = 

Similarly, U₁₃ = +5.4 × 10⁻³J and U₂₃ = −13.5 × 10⁻³ J

The total potential energy is therefore U = −1.41 × 10⁻² J

This negative potential energy means that external work is needed to separate the particles and place them at infinity.