GATE 2021 PH – Question 22
For a magnetic material, let $\mathbf H$ be magnetic field intensity, $\mathbf M$ magnetization, $\mathbf B$ magnetic flux density and $\mathbf J_b$ bound current density. Which relations are always valid, including at its surface?
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Correct answer: (B) $\nabla\cdot\mathbf B=0$; (D) $\nabla\cdot\mathbf J_b=0$
Explanation
Gauss’s law for magnetism gives $\nabla\cdot\mathbf B=0$ universally. Since $\mathbf B=\mu_0(\mathbf H+\mathbf M)$, it only follows that $\nabla\cdot\mathbf H=-\nabla\cdot\mathbf M$; the separate divergences need not vanish, especially at an interface. Bound currents are the curl of the magnetization, with the corresponding surface current included, and divergence of a curl is zero. Therefore B and D hold.