Lenz's Law — Question 4

Question 4

A shrinking loop induces an outward magnetic force on its own boundary

A circular resistive loop of radius r is forced to shrink at rate $\dot r=-u$, where u positive, in a constant out-of-page field B. Resistance R is treated as fixed and self-inductance is negligible.

1. Find induced current and sense.

2. Find the radial force per unit circumference from the applied field and identify whether it aids the shrinkage.

Question 4 - Solution

A shrinking loop induces an outward magnetic force on its own boundary

(a) Method and calculation

$$\dot\Phi=2\pi Br\dot r=-2\pi Bru.$$$$\boxed{i=2\pi Bru/R>0\quad\text{counterclockwise}}.$$

(b) Interpretation and further calculation

$$\frac{d\vec F}{d\ell}=i\hat\phi\times B\hat z=iB\hat r.$$$$\boxed{dF/d\ell=2\pi B^2ru/R\quad\text{outward}}.$$

Physical check

The magnetic force resists the imposed contraction, consistent with the flux-change rule.