# Magnetic field in matter.

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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?

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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. 