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GATE 2024 PH – Question 20

Electromagnetic theory · magnetic materials · 1 mark · Multiple choice

An infinite cylinder of radius $R$ has frozen-in magnetization $\mathbf M=ke^{-s}\hat z$, with $s$ the distance from its axis. There is no free current. With vacuum permeability $\mu_0$, the magnetic flux density inside is

  1. 0
  2. $\mu_0ke^{-R}\hat z$
  3. $\mu_0ke^{-s}\hat z$
  4. $\mu_0ke^{-s}(R/s)\hat z$

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Correct answer: (C) $\mu_0ke^{-s}\hat z$

Explanation

For a magnetised material, the fields satisfy $\mathbf B=\mu_0(\mathbf H+\mathbf M)$.

**Step 1: field equations.** With no free current, $\nabla\times\mathbf H=0$ and $\nabla\cdot\mathbf B=0$.

**Step 2: divergence of $\mathbf M$.** The magnetisation $\mathbf M=ke^{-s}\hat z$ points along the cylinder axis and depends only on the distance $s$ from the axis, so $\nabla\cdot\mathbf M=\partial M_z/\partial z=0$. In an infinite cylinder there are no end faces where magnetic "surface charge" could appear. Hence $\nabla\cdot\mathbf H=-\nabla\cdot\mathbf M=0$ and $\nabla\times\mathbf H=0$ everywhere inside, and $\mathbf H$ vanishes at infinity: $\mathbf H=0$.

**Step 3: field inside.**
$$\mathbf B=\mu_0\mathbf M=\mu_0ke^{-s}\,\hat z\quad(\text{option C}).$$