Ampère's Law — Question 7

Question 7

A thick current slab has a linear interior field and constant exterior fields

An infinite slab occupies $-a\le z\le a$ and carries uniform volume current density $J\hat x$. There is no externally imposed field. The slab is infinite along x and y. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Derive the magnetic-field vector inside and outside the slab.

2. Find the root-mean-square field across its thickness and compare it with the surface magnitude.

Question 7 - Solution

A thick current slab has a linear interior field and constant exterior fields

(a) Method and calculation

$$2B(z)\ell=\mu_0J(2|z|\ell)\Rightarrow\vec B=-\mu_0Jz\hat y\quad(|z|\le a).$$$$\boxed{\vec B=-\mu_0Ja\,\operatorname{sgn}(z)\hat y\quad(|z|\ge a)}.$$

(b) Interpretation and further calculation

$$B_{\mathrm{rms}}^2=\frac1{2a}\int_{-a}^a(\mu_0Jz)^2dz=\frac{(\mu_0Ja)^2}{3}.$$$$\boxed{B_{\mathrm{rms}}=B_{\mathrm{surface}}/\sqrt3}.$$

Physical check

The signed average field across the slab is zero by antisymmetry, while the root-mean-square field is positive.