Electric Potential Energy of a System of Point Charges — Question 3

Question 3

Removing one corner charge extracts three specific pair contributions

Four equal positive point charges q occupy square vertices A,B,C,D of side a. Other charges remain fixed when one is moved; use zero interaction energy at infinite separation.

1. Find the full initial interaction energy.

2. Find the quasistatic work removing A to infinity and the energy of the three charges left behind.

Question 3 - Solution

Removing one corner charge extracts three specific pair contributions

(a) Method and calculation

$$U_i=kq^2\left(\frac4a+\frac2{\sqrt2a}\right)=\frac{kq^2}{a}(4+\sqrt2).$$$$qV_{\mathrm{others}}(A)=\frac{kq^2}{a}\left(2+\frac1{\sqrt2}\right).$$

(b) Interpretation and further calculation

$$W_{\mathrm{ext}}=\Delta U=-\frac{kq^2}{a}\left(2+\frac1{\sqrt2}\right).$$$$U_f=U_i+W_{\mathrm{ext}}=\frac{kq^2}{a}\left(2+\frac1{\sqrt2}\right).$$

Physical check

Removing one charge eliminates its three pairs; using one half of q times the other-source potential would undercount the removed energy.