Induction and Energy Transfers — Question 8

Question 8

Crossing a wider uniform field region does not necessarily increase the heat

A rectangular loop of width w and height h is pulled at constant speed v completely across a static perpendicular field strip of width D>w. The uniform field is B, resistance R is constant and self-inductance is negligible.

1. Find the total heat produced during entry and exit.

2. Find total external mechanical work and explain its dependence on D.

Question 8 - Solution

Crossing a wider uniform field region does not necessarily increase the heat

(a) Method and calculation

$$|\mathcal E|=Bhv\quad\text{during two intervals of duration }w/v.$$$$\boxed{H=2\frac{(Bhv)^2}{R}\frac wv=\frac{2B^2h^2wv}{R}}.$$

(b) Interpretation and further calculation

$$\Delta K=0\Rightarrow\boxed{W_{\mathrm{ext}}=H}.$$$$\text{fully inside:}\quad\dot\Phi=0\Rightarrow i=0\Rightarrow P_R=0.$$

Physical check

Increasing D extends the zero-current interval. It does not change the heat from the two field boundaries in this ideal model.