B Field Inside of Sphere using Sep. Variables

[bowlbase]

Done editing I hope.

Homework Statement

If J_f = 0 everywhere, then (as we showed in class), one can express H as the gradient of a scalar potential, W. W satisfies Poisson’s equation with ∇⋅M as the source. Use this fact to find the field inside a uniformly magnetized sphere. (Griffiths has some additional verbiage
intended to help, but I think you already know what he says.) Compare your answer with
example 6.1 (p. 264-5), which is this problem solved by another method.

Homework Equations

$H^\perp_{above} - H^\perp_{below}=-(M^\perp_{above}- M^\perp_{below})$

The Attempt at a Solution

My question is with the constraints and in particular the one I have in the equations section. I had this as $∇W^\perp_{in} - ∇W^\perp_{out}=M^\perp$ Since at r=R they are equivalent. I know that $M^\perp$ must writable as some version of M but I cannot determine what. I know the solutions manual has $M\hat{z}\hat{r}= Mcos(θ)$ but I don't understand how they have $\hat{z}$ in spherical coordinates...

Thanks for any clarification.

 

[marcusl]

Can you be more precise about your confusion regarding $\hat{z}$?

 

[bowlbase]

More than the z vector I'm just confused how they got from M orthogonal to Mcos(θ). What is the reasoning or the process?

 

[marcusl]

I don't own a copy of Griffiths and you haven't shown the whole problem, so I'm going to make a guess that the magnetization is along the z axis? The component of M normal to the boundary (at r=R) is proportional to cos(theta). You can see this intuitively: at theta=0, M is normal to the boundary so Mnormal=M. At 90 deg, M is tangential to the circle so Mnormal=0.

 

[paranormal]

Dear Student,

Thank you for your question. Based on the given information, it seems that you are working on finding the magnetic field inside a uniformly magnetized sphere using the method of separation of variables. As per the problem statement, Jf=0 everywhere, which implies that the magnetic field is a conservative field and can be expressed as the gradient of a scalar potential, W. The potential satisfies Poisson's equation with ∇⋅M as the source.

To find the field inside the sphere, we can use the fact that the tangential components of the magnetic field are continuous across the boundary, i.e., $H^\perp_{above} - H^\perp_{below}=-(M^\perp_{above}- M^\perp_{below})$. We can also use the boundary conditions at r=R, which is the radius of the sphere, to determine the expressions for H and M.

In spherical coordinates, the unit vector in the z-direction is given by $\hat{z}=(sin(θ)cos(φ),sin(θ)sin(φ),cos(θ))$. Therefore, we can express the magnetization vector M as $M=M\hat{z}=(Msin(θ)cos(φ),Msin(θ)sin(φ),Mcos(θ))$. Using this expression for M, we can determine the tangential components of the magnetic field and compare it with the solution given in example 6.1 in the textbook by Griffiths.

I hope this helps clarify your doubts. If you have any further questions, please feel free to ask.

Best regards,