Electric Potential Energy of a System of Point Charges — Question 9

Question 9

Splitting a fixed charge maximizes the pair interaction at equal shares

Two point-charge sites a distance d apart carry nonnegative charges x and Q-x, with Q>0 fixed. Consider only their mutual interaction energy; exclude intrinsic self-energies and any charge-transfer apparatus.

1. Find the value of x that maximizes the mutual energy and the values that minimize it.

2. Let $x=Q/2+\delta q$. Find the exact energy change from equal sharing and explain why this calculation alone cannot predict charge sharing between conducting objects.

Question 9 - Solution

Splitting a fixed charge maximizes the pair interaction at equal shares

(a) Method and calculation

$$U(x)=\frac{k}{d}x(Q-x)=\frac{kQ^2}{4d}-\frac{k}{d}(x-Q/2)^2.$$$$U_{\max}=kQ^2/(4d)\text{ at }x=Q/2;\quad U_{\min}=0\text{ at }x=0,Q.$$

(b) Interpretation and further calculation

$$\Delta U=-\frac{k(\delta q)^2}{d}\le0.$$$$U_{\mathrm{physical}}=U_{\mathrm{mutual}}+U_{\mathrm{self}}+U_{\mathrm{apparatus}}.$$Terms excluded by this idealized problem can determine the actual equilibrium.

Physical check

Point-charge mutual energy is useful for separated particles but is not the complete charging energy of finite conductors.