Electric Potential Energy of a System of Point Charges — Question 5

Question 5

Uniform dilation rescales every pair energy by the same factor

A collection of fixed charges has initial mutual energy U0. All position vectors relative to a chosen center are multiplied quasistatically by s>0 while the charge values stay fixed.

1. Find U(s) and the external work from s=1 to s=2.

2. Find the generalized electric force conjugate to dimensionless s and discuss the sign of the work when U0 is negative.

Question 5 - Solution

Uniform dilation rescales every pair energy by the same factor

(a) Method and calculation

$$r_{ij}(s)=s r_{ij}(1)\Rightarrow U(s)=\sum_{i<j}\frac{kq_iq_j}{s r_{ij}(1)}=U_0/s.$$$$W_{\mathrm{ext}}(1\to2)=U_0/2-U_0=-U_0/2.$$

(b) Interpretation and further calculation

$$\mathcal F_s=-\frac{dU}{ds}=U_0/s^2,\quad dW_e=\mathcal F_s\,ds.$$$$U_0<0\Rightarrow W_{\mathrm{ext}}>0.$$The generalized force has energy units because s is dimensionless.

Physical check

The dilation identity holds for any geometry with Coulomb pair interactions, but it does not determine each individual force vector.