Lenz's Law — Question 3

Question 3

Local magnetic charge separation can coexist with zero loop emf

A rigid rectangular conducting loop translates at velocity $v\hat x$ entirely within uniform steady field $-B\hat z$. Its vertical sides each have length ell. Ignore resistance-dependent transients.

1. Find the magnetic force per unit positive charge along each vertical side and the open-circuit top-minus-bottom voltage.

2. Find the total motional emf around the closed loop and explain the cancellation.

Question 3 - Solution

Local magnetic charge separation can coexist with zero loop emf

(a) Method and calculation

$$\vec v\times\vec B=vB\hat y,\quad\vec E=-\vec v\times\vec B=-vB\hat y.$$$$V_{\mathrm{top}}-V_{\mathrm{bottom}}=-\int_{\mathrm{bottom}}^{\mathrm{top}}\vec E\cdot d\vec\ell=\boxed{Bv\ell}.$$

(b) Interpretation and further calculation

$$\oint(\vec v\times\vec B)\cdot d\vec\ell=Bv\ell-Bv\ell=0.$$$$\boxed{\mathcal E=0,\quad\dot\Phi=0}.$$

Physical check

The two vertical sides have the same physical polarity but are traversed in opposite directions around the contour.