Faraday's Law of Induction — Question 8

Question 8

A loop crossing a finite field strip generates separated entry and exit pulses

A rectangular loop of width w along x and height h moves right at speed v through a static field strip of width D>w. The uniform field B is out of the page. At t=0 the leading edge enters; positive circulation is counterclockwise.

1. Give the emf during entry, full immersion and exit, including interval boundaries.

2. Find the total signed charge through resistance R after the loop has completely crossed.

Question 8 - Solution

A loop crossing a finite field strip generates separated entry and exit pulses

(a) Method and calculation

$$0<t<w/v:\quad\dot A=hv\Rightarrow\mathcal E=-Bhv.$$$$w/v<t<D/v:\ \mathcal E=0;\quad D/v<t<(D+w)/v:\ \boxed{\mathcal E=+Bhv}.$$

(b) Interpretation and further calculation

$$Q_{\mathrm{entry}}=-Bhw/R,\quad Q_{\mathrm{exit}}=+Bhw/R.$$$$\boxed{Q_{\mathrm{net}}=0}.$$

Physical check

The pulse separation is set by the field-strip width; the duration of each pulse is set by the loop width.