Faraday's Law of Induction — Question 6

Question 6

A moving boundary and changing field contribute separate terms to one emf

A sliding rod of length ell encloses width $x(t)=vt$ in a uniform field into the page with magnitude $B(t)=B_0+\alpha t$. Choose clockwise circulation, so the positive normal is into the page. Start at t=0; neglect self-inductance.

1. Find flux and signed emf.

2. At t=B0/alpha, compare the full emf with the rod motional contribution alone.

Question 6 - Solution

A moving boundary and changing field contribute separate terms to one emf

(a) Method and calculation

$$\Phi=B(t)\ell x(t)=\ell v(B_0t+\alpha t^2).$$$$\boxed{\mathcal E=-\ell v(B_0+2\alpha t)}=-B\ell v-\ell x\dot B.$$

(b) Interpretation and further calculation

$$t=B_0/\alpha:\quad\mathcal E=-3B_0\ell v,\quad\mathcal E_{\mathrm{motional}}=-2B_0\ell v.$$$$\boxed{\mathcal E/\mathcal E_{\mathrm{motional}}=3/2}.$$

Physical check

The transformer and motional terms arise from the same total flux derivative; omitting either loses part of the emf.