How Do Scalar Magnetic Potentials Relate to Cylindrical Fields?

[Dassinia]

Hello,
I'm really stuck, I don't know how to start !

Homework Statement

In the regions where J_l=0, we have ∇x H=0, so we can introduce a magnetic scalar potential Vm such as H=-∇Vm. A long cylinder of radius R of linear magnetic material of permeability μ_r. The cylinder axis is in z direction, and there's a H field that is Ho*x^ far from the cylinder.
For symetrical reasons:
V_m=(As+B/s) cos(phi) V_m'=C*s*cos(phi)

V_m the potential outside the cylinder and V_m' inside.
Express A, B and C in terms of the other problem constants.

Homework Equations

∇⋅(μ_oH+μ_oM)=0 * * so * ∇⋅H=−∇⋅M
-ρ_m=∇²V_m

The Attempt at a Solution

If someone can explain me the steps to follow it would be great
I don't even understand what is the angle phi
Thanks

 

[jbsteele]

a lot</code>This problem can be solved by using two different equations. The first equation is the divergence of a vector and the second equation is Poisson's equation.Divergence: The divergence equation states that the curl of a vector is equal to zero. This means that if we take the divergence of the vector H, then it should be equal to zero in the regions where Jl = 0. This means that we can introduce a magnetic scalar potential Vm such that = -∇VmPoisson's equation: Poisson's equation states that the laplacian of a function is equal to the negative product of the density and the function. For this problem, the density is the magnetic material permeability μr and the function is the magnetic scalar potential Vm. So, we have:-ρm = ∇²VmSolution: Now that we have the two equations, we can solve for the constants A, B, and C. First, we need to determine the angle Phi. Since we are dealing with a cylinder of linear magnetic material with its axis in the z direction, Phi will be the angle between the x axis and the radial vector from the cylinder. Next, we can substitute the equation for H into the equation for the divergence of a vector and solve for Vm. We can then substitute this equation into Poisson's equation and solve for Vm'. Finally, we can use these equations to solve for A, B, and C. A = -∇Vm, B = (1/s)∇²Vm C = s∇²Vm where s is the radius of the cylinder. I hope this helps! Good luck!