Electric Potential Energy of a System of Point Charges — Question 4

Question 4

A zero total interaction energy does not imply zero forces

Charges +q, +q and -q/2 occupy vertices A,B,C of an equilateral triangle of side a. Let A=(0,0), B=(a,0), C=(a/2,$\sqrt3a/2$), with q,a>0.

1. Find the total interaction energy.

2. Find the force on A and its magnitude to test whether the system is in equilibrium.

Question 4 - Solution

A zero total interaction energy does not imply zero forces

(a) Method and calculation

$$U=\frac{k}{a}\left(q^2-\frac{q^2}{2}-\frac{q^2}{2}\right)=0.$$$$\mathbf F_{AB}=-\frac{kq^2}{a^2}\hat{\mathbf x},\quad\mathbf F_{AC}=\frac{kq^2}{2a^2}\left(\frac12\hat{\mathbf x}+\frac{\sqrt3}{2}\hat{\mathbf y}\right).$$

(b) Interpretation and further calculation

$$\mathbf F_A=\frac{kq^2}{a^2}\left(-\frac34\hat{\mathbf x}+\frac{\sqrt3}{4}\hat{\mathbf y}\right).$$$$|\mathbf F_A|=\frac{\sqrt3kq^2}{2a^2}\ne0.$$

Physical check

An energy value is a scalar at one configuration; forces depend on how energy changes under independent displacements.