Maxwell's Equations — Question 9

Question 9

A current sheet produces a tangential magnetic-field jump

An infinite sheet at z=0 carries surface current $\mathbf K=K\hat x$. Vacuum lies on both sides. Symmetry gives uniform opposite magnetic fields tangent to the sheet.

1. Use a narrow rectangular Ampere loop to find B above and below the sheet.

2. Check normal-field continuity and write the vector jump condition with normal +z.

Question 9 - Solution

A current sheet produces a tangential magnetic-field jump

(a) Method and calculation

$$2B\ell=\mu_0K\ell\Rightarrow B=\mu_0K/2.$$$$\boxed{\mathbf B_{z>0}=-\mu_0K\hat y/2,\quad\mathbf B_{z<0}=+\mu_0K\hat y/2}.$$

(b) Interpretation and further calculation

$$B_{2z}=B_{1z}=0,\quad\mathbf B_2-\mathbf B_1=-\mu_0K\hat y.$$$$\boxed{\hat z\times(\mathbf B_2-\mathbf B_1)=\mu_0\mathbf K}.$$

Physical check

The tangential jump encodes surface current, while the normal magnetic component remains continuous.