Ampère's Law — Question 6

Question 6

Two infinite current sheets confine a uniform magnetic field between them

Two infinite sheets parallel to xy lie at z=-a and z=+a. The lower carries surface current $K\hat x$ and the upper carries $-K\hat x$, with K positive. There is no applied background field. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Use a rectangular Ampère path and symmetry to find the field of one sheet.

2. Superpose the two fields to obtain B in the three regions.

Question 6 - Solution

Two infinite current sheets confine a uniform magnetic field between them

(a) Method and calculation

$$2B\ell=\mu_0K\ell\Rightarrow B_{\mathrm{sheet}}=\mu_0K/2.$$$$\vec B_{K\hat x}(z>z_s)=-\frac{\mu_0K}{2}\hat y,\quad\vec B_{K\hat x}(z<z_s)=+\frac{\mu_0K}{2}\hat y.$$

(b) Interpretation and further calculation

$$|z|<a:\quad\vec B=-\frac{\mu_0K}{2}\hat y-\frac{\mu_0K}{2}\hat y=\boxed{-\mu_0K\hat y}.$$$$\boxed{\vec B=0\quad(z>a\text{ or }z<-a)}.$$

Physical check

The opposite currents add their fields between the sheets and cancel outside; finite sheets would have edge fields.