Loops and magnet (braking force)

Problem

A magnet falls along the axis of four conducting loops, identical in area and resistance but differing in number of turns and magnet strength: (A) $N=1$, weak magnet; (B) $N=10$, weak magnet; (C) $N=10$, strong magnet ($10$ times $B$); (D) $N=100$, strong magnet. Rank from minimum to maximum braking force during the crossing.

Solution

What the braking force depends on. At equal velocity, the induced EMF is proportional to the number of turns and to the field: $\mathcal{E} \propto N B$. The induced current (at fixed resistance) is likewise $\propto N B$. The braking force arises from the interaction between this current and the field, bringing in another factor of $N B$: $F \propto (N B)^2$

Relative values of $(NB)^2$ (taking $B=1$ for the weak magnet, $B=10$ for the strong one): $\begin{aligned} \text{A:} \ (1 \cdot 1)^2 = 1 \qquad \text{B:} \ (10 \cdot 1)^2 = 100 \\ \text{C:} \ (10 \cdot 10)^2 = 10^4 \qquad \text{D:} \ (100 \cdot 10)^2 = 10^6 \end{aligned}$

Order from minimum to maximum braking force: $\ev{\text{A} < \text{B} < \text{C} < \text{D}}$

Collegamenti

Argomenti: Electromagnetic induction Concetti: Lenz’s law · Eddy currents Competenze: Ranking reasoning