Slab of material in magnetic field. Determine magnetic field

[SalcinNossle]

Homework Statement

A very large slab of material of thickness $d$ lies perpendicularly to a uniform magnetic field $\vec H_0 = \vec a_zH_0$, where $\vec a_z$ is the unit vector in the z-direction. Determine the magnetic field intensity (ignoring edge effect) in the slab:

a) if the slab material has a magnetic permeability μ

b) if the slab is a permanent magnet having magnetization vector $\vec M_i = \vec a_zM_i$2. The attempt at a solution

I really don't know how start with this. I just want some hints, would be very appreciated!

I tried looking at the boundary conditions,

$μ_1H_{1n}=μ_2H_{2n}$,

and

$\vec a_{n2}\times(\vec H_1 - \vec H_2)=\vec J_s$,

but I couldn't get anywhere.. :( Help!

 

[rude man]

a) There's a very simple equation relating B to H. What is it?
b) There's a still simple equation relating B, H and M. What is it?

 

[SalcinNossle]

$\vec H=\vec B/μ \rightarrow \vec B_0/μ=μ_0\vec a_zH_0/μ$

and

$\vec H =\vec B/μ_0- \vec M \rightarrow \vec H=μ_0\vec a_zH_0/μ_0-\vec M_i=\vec a_z(H_0-M_i)$

Thank you!

 

[rude man]

$\vec H=\vec B/μ \rightarrow \vec B_0/μ=μ_0\vec a_zH_0/μ$

and

$\vec H =\vec B/μ_0- \vec M \rightarrow \vec H=μ_0\vec a_zH_0/μ_0-\vec M_i=\vec a_z(H_0-M_i)$

Thank you!

So we have B = μ₀(H + M)
where B, H and M are all vectors, not necessarily pointing in the same direction.

In part (a) you have magnetic material but not a permanent magnet. So what is the relative direction between B and H? In other words, do they add or subtract?

In part (b) you have to analyze the permanent magnet in terms of magnetizing or Amperian currents, which changes some of the vectors' directions with respect to each other, so they may not all just add. You mentioned one boundary condition which boils down to B_1n = B_2n. But there is also a boundary condition on H. What does H do at the boundary? If you draw an integration loop which crosses the surface, you know that ∫H ds around this path = 0 since there is no actual current piercing this loop. It's a bit difficult but you can figure out what the direction of H has to do to satisfy this integral (or you can look it up! ).

BTW H, B and M are all collinear but do not necessarily point in the same direction.

 

[rezeew]

I would approach this problem by first understanding the fundamental principles of magnetism. The magnetic field is a vector quantity that describes the strength and direction of the magnetic force at any given point. In this case, the magnetic field is perpendicular to the surface of the slab, indicating that the slab is experiencing a force in the z-direction.

To determine the magnetic field intensity, we can use the equation \vec H = \frac{\vec B}{\mu_0}, where \vec B is the magnetic flux density and \mu_0 is the magnetic permeability of free space. In part a), we are given the magnetic permeability of the slab, so we can use this equation to calculate the magnetic field intensity inside the slab.

In part b), we are dealing with a permanent magnet. This means that the magnetic field is created by the alignment of atomic dipoles within the material. The magnetization vector, \vec M_i, describes the strength and direction of this internal magnetization. We can use the equation \vec B = \mu_0(\vec H + \vec M) to calculate the magnetic field intensity in this case.

To account for the slab's thickness, we can use the concept of magnetic flux density, which is the amount of magnetic field passing through a unit area. This is given by the equation \vec B = \mu_0\vec H + \mu_0\vec M.

To determine the magnetic field intensity at the edges of the slab, we can use the concept of boundary conditions. These are equations that describe the relationship between the magnetic field and magnetic flux density at the interface between two materials. In this case, we can use the equation \vec B_1\cdot\vec n_1 = \vec B_2\cdot\vec n_2, where \vec n_1 and \vec n_2 are the unit normal vectors of the two materials. This will help us account for any edge effects.

By combining these equations and principles, we can determine the magnetic field intensity inside the slab for both cases (a and b). It may also be helpful to visualize the situation and draw diagrams to aid in understanding the problem. I hope this helps and good luck with your homework!