GATE 2023 PH – Question 31
Consider an isolated magnetized sphere of radius R with a uniform magnetization M⃗⃗ along the positive z direction, with the north and south poles of the sphere lying on the z axis. It is given that the magnetic field inside the sphere is B⃗ = 2μ0 3 M⃗⃗ , where μ0 is the permeability of vacuum. Which of the following statements is(are) CORRECT?
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Correct answer: (A) The bound volume current density is zero; (B) The bound surface current reaches $|\mathbf M|$ at the equator; (D) $\mathbf m/(4\pi R^3)=B\hat z/(2\mu_0)$
Explanation
A uniformly magnetised sphere of radius $R$ with magnetisation $\mathbf M=M\hat z$ has the internal field $\mathbf B=\dfrac{2\mu_0}{3}\mathbf M$ (given).
- **A. The bound volume current density is zero.** True. $\mathbf J_b=\nabla\times\mathbf M=0$ for uniform $\mathbf M$. ✓
- **B. The bound surface current reaches $|\mathbf M|$ at the equator.** True. $\mathbf K_b=\mathbf M\times\hat n=M\sin\theta\,\hat\phi$, which is largest at $\theta=90^\circ$ (the equator), where it equals $M$. ✓
- **C. $\mathbf H=-2\mathbf M/3$.** False. $\mathbf H=\dfrac{\mathbf B}{\mu_0}-\mathbf M=\dfrac23\mathbf M-\mathbf M=-\dfrac13\mathbf M$.
- **D. $\dfrac{\mathbf m}{4\pi R^3}=\dfrac{B\hat z}{2\mu_0}$.** The dipole moment is $\mathbf m=\dfrac43\pi R^3\mathbf M$, so $\dfrac{m}{4\pi R^3}=\dfrac M3$. Also $\dfrac B{2\mu_0}=\dfrac{(2\mu_0M/3)}{2\mu_0}=\dfrac M3$. They are equal. ✓
Answer **A, B and D**.