Faraday's Law of Induction — Question 2

Question 2

A spatial field gradient must be integrated before its time derivative is taken

A stationary rectangular N-turn coil spans $0<x<a$ and $0<y<b$. Its normal is +z and $B_z(x,t)=\beta t(1+x/a)$.

1. Find its flux linkage and induced emf.

2. Compare the exact emf with using the field at the left edge and at the midpoint.

Question 2 - Solution

A spatial field gradient must be integrated before its time derivative is taken

(a) Method and calculation

$$\Lambda=N\int_0^b\!dy\int_0^a\beta t(1+x/a)dx=\frac32Nab\beta t.$$$$\boxed{\mathcal E=-\frac32Nab\beta}.$$

(b) Interpretation and further calculation

$$\mathcal E_{\mathrm{left}}=-Nab\beta=\frac23\mathcal E.$$$$B_z(a/2,t)=\frac32\beta t\Rightarrow\boxed{\mathcal E_{\mathrm{midpoint}}=\mathcal E}.$$

Physical check

The midpoint works exactly for a linear spatial profile, not for arbitrary nonuniform fields.