Calculating the Potential from the Field — Question 5

Question 5

A screened radial field integrates by recognizing a total derivative

For r>0 a prescribed field is $E_r=C e^{-r/\ell}(r^{-2}+1/(\ell r))$, where C and $\ell$ are positive. Set V(infinity)=0.

1. Find V(r) and verify it by differentiation.

2. Find the quasistatic external work moving charge q from r=$\ell$ to r=$2\ell$.

Question 5 - Solution

A screened radial field integrates by recognizing a total derivative

(a) Method and calculation

$$\frac d{dr}\left(\frac{e^{-r/\ell}}r\right)=-e^{-r/\ell}\left(\frac1{r^2}+\frac1{\ell r}\right).$$$$V(r)=\frac{C e^{-r/\ell}}r,\quad -V\prime(r)=E_r.$$

(b) Interpretation and further calculation

$$\Delta V=\frac C\ell\left(\frac{e^{-2}}2-e^{-1}\right).$$$$W_{\mathrm{ext}}=q\Delta V\approx-.300212\frac{qC}{\ell}.$$

Physical check

The infinity reference fixes the integration constant; for positive q the external agent extracts energy during outward motion.