How Does Magnetic Field Continuity Affect Scalar Potential in Linear Materials?

[quasar987]

The problem: In the regions of space where $\vec{J_f} = 0$, the curl of $\vec{H}$ vanishes, and hence we can define a scalar potential $V_m$.

(a) Show that $V_m$ must be continuous at the boundary of material. - Done

(b) Consider a very long cylinder of radius a made out of a linear magnetic material of relative permeability $\mu_r$. The axis of the cylinder is oriented along $\hat{z}$ and the cylinder is emerged in a field $\vec{H}$ that is worth $H_0 \hat{x}$ very far from it. From symetrical considerations, $V_m$ must be of the form

$$V_{m_1}=(As+B/s)cos\phi$$
$$V_{m_2}=Cscos\phi$$

Where $V_{m1}$ is the potential outside the cylinder and $V_{m2}$ the one inside. Find the value of the constant A, B and C in terms of the other parameters.


My solution: I used the condition of continuity to find B in terms of C, and I used the condition at infinity to find A = -H_0.

What is the 3rd condition on V that'll let me find the value of the third constant?

 

[quasar987]

Nerver mind, I found it. It was crazy! It had to do with the boundary conditions on H, expressed as the gradient of V and noticing that in the cylinder along phi = pi/2, the H field is purely solenoidal, then so is the magnetization, which implied that the dot product of the gradient of V outside in the limit s-->a with the normal unit vector is 0, which allowed to recover a second relation btw B and C.

Second Edit: Wrong again! OMG it doesn't end. Finally, I got the right thing. The third relation was lying not to far below the boundary condition on H perpendicular. Much less complicated than what I previcously thought but also much less fun.