Boundary conditions ##\vec{B}## and ##\vec{H}##

happyparticle
Homework Statement:
Find boundary conditions ##\vec{B}## and ##\vec{H}## for a cylinder of radius a and length 4a and ##\vec{M} = M\hat{z}## on the axis of the cylinder
Relevant Equations:
##\vec{\nabla} \cdot \vec{B} = 0##
##\vec{\nabla} \cdot \vec{H} = - \vec{\nabla} \cdot \vec{M}##
##\vec{\nabla} x \vec{B} = \mu_0 \vec{J}##
##\vec{\nabla} x \vec{H} = \mu_0 \vec{J}_f##
When asking for boundary conditions I'm wondering if this is enough in this situation to give
##\vec{\nabla} \cdot \vec{B} = 0 , B_{2\perp} - B_{1 \perp} = 0##
##\vec{\nabla} \cdot \vec{H} = - \vec{\nabla} \cdot \vec{M}, H_{2\perp} - H_{1 \perp} = - (M_{2\perp} - M_{1 \perp})##
##\vec{\nabla} \times \vec{B} = \mu_0 \vec{J}, B_{2\||} - B_{1 \||} = \mu_0 \vec{K} \times \hat{n}##
##\vec{\nabla} \times \vec{H} = \mu_0 \vec{J}_f , H_{2 \||} - H_{1 \||} = \vec{K}_f \times \hat{n}##

Homework Helper
Gold Member
It probably isn't necessary to include anything about the curls of the vectors, because that is a surface current type treatment that is completely separate from the pole model of magnetostatics. Meanwhile, what do you know about ## M_{2 \, perpendicular} ## ? =if it is outside of the material? For the ## M_{in \, perpendicular} ##, it is ## -M ## on the left endface, and ## +M ## on the right endface.

Edit: Scratch part of that=I think they may be looking for boundary conditions everywhere on the surface, (they aren't completely clear here=do they want just the on-axis conditions? ), and it may be useful to employ the surface current model to get the conditions for the parallel components.

It may be worth mentioning that this problem has two very standard ways of solving it, and Legendre's method with boundary conditions are not needed to solve it. See https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/

Last edited:
happyparticle
Homework Helper
Gold Member
a follow-on: It still needs a little work, but let's get started using ## \nabla \times \vec{H}=\vec{J}_{free}=0 ##. Using Stokes theorem on this, what can you say about the parallel components of ## \vec{H} ## anywhere on the entire surface, i.e. ## H_{out \, parallel} ## and ## H_{in \, parallel} ##?

happyparticle
Finally find it, thank you.