Lenz's Law — Question 7

Question 7

The current reverses at maximum linkage while the magnetic drag retains its direction

A small resistive pickup moves at constant positive speed v along x through a localized source with linkage $\Lambda(x)=\Lambda_0e^{-x^2/a^2}$. Neglect self-inductance. Use the supplied mechanical force relation $F_x=i\,d\Lambda/dx$.

1. Find current and its sign on either side of x=0.

2. Find force and show it opposes motion on both sides.

Question 7 - Solution

The current reverses at maximum linkage while the magnetic drag retains its direction

(a) Method and calculation

$$\Lambda\prime(x)=-\frac{2x}{a^2}\Lambda(x),\quad i=-\frac vR\Lambda\prime(x).$$$$\boxed{i=\frac{2vx\Lambda(x)}{Ra^2}}:\quad i<0\ (x<0),\ i>0\ (x>0).$$

(b) Interpretation and further calculation

$$F_x=i\Lambda\prime=-\frac vR[\Lambda\prime(x)]^2.$$$$\boxed{F_x=-\frac{4vx^2\Lambda(x)^2}{Ra^4}\le0}.$$

Physical check

The current reverses because the linkage slope reverses. Drag depends on the square of that slope and does not reverse.