Constant magnetic field > The magnetic field from a current distributed over a surface of space > Problem 9.3.14

Statement

$9.3.14.$ A disk of radius $R$ and height $h ≪ R$, made of a material with magnetic permeability $µ = 1 + κ$, $κ ≪ 1$,
was placed across a uniform magnetic field of induction $B_0$. How much will the induction in the center
of the disk differ from $B_0$?

Solution

Under the influence of an external field, a magnetic moment of magnitude is induced in a paramagnet $M=\chi H$,where $H$ -
magnetic intensity associated with $B$
by ratio $B = \mu \mu_0 H$,
due to small size of $\chi$, $$M\approx \frac{B_0\chi}{\mu_0}$$

As we know, the disk field at its center can be replaced by the ring current field $I = Mh$,
which is equal to $$B=\frac{\mu_0 I}{2R}$$, this is the desired change in the field, we substitute:$$\Delta B=\frac{B_0\chi h}{2R}$$

Answer

$$\Delta B=\frac{B_0\chi h}{2R}$$