Cylinder parallel to a constant external B field

[sayebms]

Homework Statement

A cylinder of permeability ##\mu## is placed in an external field ##B_0##. find the strength and direction of magnetic field inside the cylinder for:
a) when axis of cylinder is parallel to external field.
b) when axis of cylinder makes an angle ##\theta _0## with external field.

Homework Equations

Conditions of magneto statics:$$\nabla \times H =0\\H=-\nabla{\phi}\\\nabla^2{\phi}=0$$
the Laplace operator is in cylindrical coordinates

The Attempt at a Solution

for part a I take B to be in x direction. its clear from the problem that there is a symmetry in the ##\phi## direction so this means that the magnetic scalar potential does not depend on ##\phi##. hence the Laplace equation reduces to $$\frac{1}{\rho}\frac{\partial}{\partial{\rho}}(\rho \frac{\partial \phi}{\partial{\rho}})+\frac{\partial^2\phi}{\partial z^2}=0$$ but here is the problem; I know that the magnetic field outside and far from cylinder must be a constant (##B_0##) which means that potential must have a term first order in ##z## but as we see from the Laplace equation we don't get such a thing. what am I doing wrong here? is this even the right approach for solving this problem? [/B]

 

[mark726]

Your approach is correct, but you are missing a boundary condition. In order to determine the magnetic scalar potential, you need to specify the boundary conditions at the surface of the cylinder. In this case, the boundary condition is that the tangential component of the magnetic field (i.e. the component parallel to the surface of the cylinder) must be continuous across the surface. This means that the magnetic scalar potential must satisfy the equation $$\frac{\partial \phi}{\partial z}\bigg|_{z=0} = \frac{\partial \phi}{\partial z}\bigg|_{z=h}$$ where ##h## is the height of the cylinder. This condition will help you determine the first-order term in ##z## that you are missing. Once you have that, you can solve the Laplace equation to find the magnetic scalar potential and then use it to determine the magnetic field inside the cylinder.