Magnetic field in matter.

[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.

 

[quicknote]

The third condition on V that will allow you to find the value of the third constant is the boundary condition at the surface of the cylinder. This condition states that the tangential component of the magnetic field must be continuous across the boundary. This can be written as:

\mu_r H_{0}\cos\theta = H_{0}\cos\theta + B\sin\theta - C\sin\theta

Where \theta is the angle between the axis of the cylinder and the direction of the external magnetic field. This equation can be simplified to:

(\mu_r -1)H_{0}\cos\theta = (B-C)\sin\theta

Therefore, the value of the third constant, C, can be found by setting \mu_r = 1 and solving for C. This gives us:

C = B - (\mu_r - 1)H_{0}\cot\theta

Substituting this value of C into the expression for V_{m2}, we can find the value of B in terms of the other parameters:

B = \frac{C}{\cos\phi} = \frac{B}{\cos\phi} - (\mu_r - 1)H_{0}\cot\theta \csc\phi

Therefore, the values of all three constants, A, B, and C, can be determined in terms of the other parameters in the problem. This allows us to fully characterize the potential V_m and understand the behavior of the magnetic field inside and outside the cylinder.