What is the $\mathbf{H}$-field inside and outside the magnet?

Question

A permanent cylindrical magnet with diameter $D$ and length $L$ possesses an uniform magnetization $\mathbf{M}$ going in the $z$-direction in a cylindrical coordinate system. The magnet is placed with its centre in the origin, with its length going along the $z$-axis.

Problem

Given the expression for the magnetic field along the $z$-axis $$\mathbf{B}(z)= \mathbf{a_z} \cdot \frac{\mu_0M}{2} \Bigg[ \frac{z+ \frac{L}{2}}{\sqrt{\big(z+\frac{L}{2} \big)^2+ \big(\frac{D}{2} \big)^2}}- \frac{z- \frac{L}{2}}{\sqrt{\big(z-\frac{L}{2} \big)^2+ \big(\frac{D}{2} \big)^2}} \Bigg] $$ arrive at an expression for the $\mathbf{H}$-field along the $z$-axis. What is the direction of $\mathbf{H}$ inside and outside the magnet?

My attempt

Given the simple relation $\mathbf{H}=\frac{\mathbf{B}}{\mu_0}-\mathbf{M}$ we get

$$\mathbf{H}(z)=\mathbf{a_z} \cdot \frac{M}{2} \Bigg[ \frac{z+ \frac{L}{2}}{\sqrt{\big(z+\frac{L}{2} \big)^2+ \big(\frac{D}{2} \big)^2}}- \frac{z- \frac{L}{2}}{\sqrt{\big(z-\frac{L}{2} \big)^2+ \big(\frac{D}{2} \big)^2}} \Bigg]-M \cdot \mathbf{a_z} $$

But I'm not sure how to answer what direction $\mathbf{H}$ has inside and outside the magnet.

Edit

I found this example in my book, that revolves around the exact same scenario, a cylindrical magnet with uniform magnetization $\mathbf{a_z}M_0$. In the example, the author writes about the relationship about $\mathbf{H}$ and $\mathbf{B}$. He also talks about their direction, which I have marked with yellow.

So if the scenario is the same, doesn't the example answer also answer my problem completely?

Answer 1

I will give a more qualitative answer than just formulas. Considering the given proposal one has to be very careful since $\mathbf{H}$ not only has to fulfill Maxwell-equations and material equations, but also the boundary conditions which are not the same for $\mathbf{B}$ and $\mathbf{H}$.

The Maxwell equations in case of a permanent magnet without any currents are those of magnetostatics:

$\nabla \times \mathbf{H}= 0 \quad\quad \text{and} \quad\quad \nabla\cdot \mathbf{B}=0$

and there is the relevant material equation (assuming cgi-units): $\mathbf{B} = \mathbf{H} + 4\pi \mathbf{M}$.

As we want to know $\mathbf{H}$ we eliminate $\mathbf{B}$ and get:

$$\nabla \cdot \mathbf{H} = -4\pi \nabla \cdot\mathbf{M} \quad {and} \quad \nabla \times \mathbf{H}= 0$$

This system of equations resembles to the equations of electrostatics:

$$ \nabla \cdot \mathbf{E} = 4\pi \rho \quad {and} \quad \nabla \times \mathbf{E}= 0$$

So the solution will resemble an already well-known case of electrostatics.

In the following we also consider that the permanent magnet is directed along the z-axis, the only non-zero component of the magnetisation is $M_z$.

The only locations where the divergence of the magnetisation is non-zero are the ends of the permanent magnet. At the borders along the magnet, the z-component of the magnetisation $M_z$ also suffers an abrupt change, but in radial direction, so it does not matter since changes of $M_z$ in radial direction do not contribute to the divergence (only $\frac{\partial M_z}{\partial z}$ contributes).

So the permanent magnet resembles to two plates of opposite electrical charges at each end of the magnet, i.e. the $\mathbf{H}$-field looks like that of a cylindrical electrical condensator.

The equivalent charge of such a system is: $\rho = -\frac{\partial M_z}{\partial z}$ and assuming that the abrupt change at each end of the magnet is the same (respectively symmetrical) both charge densities $\rho_1 = - \rho_2$ as it is for a condensator. If the diameter $D \rightarrow 0$, the field line picture would resemble to that of two opposed point charges $+$ and $-$. So in order to know the $\mathbf{H}$ field in space as a function of coordinates just refer to the $\mathbf{E}$ field in space for a condensator.