A conducting shield for fluctuating magnetic fields
It is often necessary to shield instruments from stray magnetic fields. If the only disturbing field is that of the earth, then one can set up a pair of Helmholtz coils (Prob. 18-9) to oppose the earth's field. If the field is static but not uniform, then one must use a shield made of high-permeability material. Multiple shields, one inside the other, are better than a single thick shield.
Could a conducting enclosure be a good shield against fluctuating magnetic ficlds? The answer is yes, as we shall see, but only at quite high frequencies.
Imagine a simple situation where the external magnetic field $B_{e x}$ is uniform, with $B_{\mathrm{rz}}=B_{\text {es.m }}$ exp jot. The shield is a long tube, parallel to the lines of $\boldsymbol{B}$, a few times longer than its diameter $2 a$, and a few times longer than the shiclded region.
We assume that the current induced in the shield is uniformly distributed, throughout its thickness $b$. In other words, we disregard the skin effect, (Sec. 29.1). If this assumption is not valid, then the shielding is better than our calculation would indicate.
(a) Calculate the resistance $R$ ' of the tube, per unit length, in the azimuthal direction. Calculate $L$.
(b) Let $B_{m}=B_{\text {in, in }}$ exp jot be the value of $B$ inside the tube, away from the ends. Find the ratio $B_{\text {in. }} / B_{e x}, m$
(c) Show that, if the skin effect is negligible, this ratio cannot be smaller than $0.5$. A conducting enclosure therefore acts as a shield only through the skin effect.