Calculating the Potential from the Field — Question 4

Question 4

A radial power law admits an infinity reference only for sufficiently fast decay

A radial field outside r=a is $\mathbf E=C r^{-n}\hat{\mathbf r}$ with C positive and n real.

1. Determine for which n the convention V(infinity)=0 is possible and derive V(r) in that case.

2. For n=1 use V(a)=0 instead; find V(r) and the voltage from a to 3a.

Question 4 - Solution

A radial power law admits an infinity reference only for sufficiently fast decay

(a) Method and calculation

$$V(r)=\int_r^\infty C u^{-n}\,du.$$$$n>1:\quad V(r)=\frac{C}{n-1}r^{1-n};\qquad n\le1:\ \text{integral diverges}.$$

(b) Interpretation and further calculation

$$n=1:\quad V(r)-V(a)=-\int_a^r\frac C u\,du=-C\ln(r/a).$$$$V(3a)-V(a)=-C\ln3.$$

Physical check

Finite voltage differences can exist even when choosing zero potential at infinity is mathematically impossible.