Demagnetizing factor. For a strongly magnetic material, the internal $B_i$ field may differ considerably from $\mu_0 H_1$. One has $M=\chi H_i$ and
$$
H_i=H_{0 I}-\alpha M
$$
where $\alpha$ is called the demagnetizing factor, its value depends on the shape of the sample. For a long cylindrical sample, $\alpha=0$; for a sphere, $\alpha=$ $1 / 3$.
(i) Show that $H_i=H_{0 I} /(1+\alpha \chi)$ and hence
$$
M=\frac{\chi}{1+\alpha \chi} H_{0 I} .
$$
This means that the influence of the magnetization on the field can be accounted for in many expressions by replacing the susceptibility by $\chi /(1+\alpha \chi)$, but this simple rule is not universally true because we still have $B_i=\mu_0(1+\chi) H_i$.
(ii) Show that the generalization of $(14.20)$ is
$$
B_0=B_{0 I}\left(1+\gamma_0 \frac{(1-\alpha) \chi}{1+\alpha \chi}\right),
$$
where $\gamma_0$ is the geometric factor defined in question 14.9. Note, since $\alpha$ also depends on the geometry, it is common to absorb the factor $(1-\alpha)$ into the geometric factor, such that one has
$$
B_0=B_{0 I}\left(1+\gamma \frac{\chi}{1+\alpha \chi}\right),
$$
where $\gamma=(1-\alpha) \gamma_0$.