Electric Potential Energy of a System of Point Charges — Question 10

Question 10

Changing one source charge alters only its incident pair energies

Four fixed charges +q occupy square vertices A,B,C,D of side a. An external process changes the charge at A continuously from +q to -q, leaving all positions and other charges fixed. Track only mutual interaction energy.

1. Find the change in mutual energy using the potential at A from the other three sources.

2. Find the final total mutual energy and verify it by direct pair counting.

Question 10 - Solution

Changing one source charge alters only its incident pair energies

(a) Method and calculation

$$V_{\mathrm{others}}(A)=\frac{kq}{a}\left(2+\frac1{\sqrt2}\right).$$$$\Delta U=\int_q^{-q}V_{\mathrm{others}}\,dq_A=-\frac{2kq^2}{a}\left(2+\frac1{\sqrt2}\right).$$

(b) Interpretation and further calculation

$$U_f=\frac{kq^2}{a}(4+\sqrt2)-\frac{kq^2}{a}(4+\sqrt2)=0.$$$$U_{\mathrm{sides}}=(-1-1+1+1)\frac{kq^2}{a}=0,\quad U_{\mathrm{diagonals}}=(-1+1)\frac{kq^2}{\sqrt2a}=0.$$

Physical check

The result is a change of mutual electrostatic energy, not a claim about the total work or chemical energy of the charge-transfer mechanism.