Induced Electric Fields — Question 4

Question 4

Two paths between the same points can give different electric work

Inside a uniform cylindrical field increasing at rate beta, the induced field is $E_\phi=-\beta r/2$. A charge q is moved quasistatically from (r,0) to (-r,0), either on the upper semicircle counterclockwise or on the lower semicircle clockwise. Treat beta as constant.

1. Find the work done by the induced electric field along each path.

2. Find their difference and compare with a full-loop circulation.

Question 4 - Solution

Two paths between the same points can give different electric work

(a) Method and calculation

$$W_{\mathrm{upper}}=q\int_0^\pi E_\phi r\,d\phi=\boxed{-q\pi\beta r^2/2}.$$$$W_{\mathrm{lower}}=q\int_0^{-\pi}E_\phi r\,d\phi=\boxed{+q\pi\beta r^2/2}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{upper}}-W_{\mathrm{lower}}=-q\pi\beta r^2.$$$$\boxed{W_{\mathrm{upper}}-W_{\mathrm{lower}}=q\oint\vec E\cdot d\vec\ell=-q\dot\Phi}.$$

Physical check

A single-valued electrostatic potential cannot represent this induced field around the entire loop.