# Why does the Magnetic Flux Density B, saturates in a ferromagnetic material with increasing H?

I understand that the magnetization must saturate as more and more domains are aligned. But $B$ is still directly proportional to $H$, and hence it must increase linearly with $H$. But every book that teaches $B-H$ curve, says that it saturates after some time. How can this be so?

## 1 Answer

No, your understanding is wrong: $B$ isn't proportional to $H$, the relationship is $\vec{B}=\mu_0 (\vec{H} + \vec{M}(\vec{H}))$ where $\vec{M}$ is the magnetization (see Wiki page of this name). And $\vec{M}$ saturates for precisely the reason you state: its maximum value is reached when all the magnetic dipoles in a medium are perfectly aligned with the ambient field $\vec{H}$. At very high fields, when the material saturates, $\vec{B}$ keeps increasing with increasing $H$ owing to the "freespace" $\mu_0\vec{H}$ term in $\vec{B}=\mu_0 (\vec{H} + \vec{M}(\vec{H}))$, but for many materials, the gradient $\mu_0$ is extremely small compared to the rate of change of $\vec{M}$ in the unsaturated region, so the $B-H$ curve looks as though it flattens out even though its gradient has fallen to $\mu_0\ll \left.\mathrm{d}_H M\right|_{H=0}$.