GATE 2024 EE – Question 52
The given equation represents a magnetic field strength $\bar H(r,\theta,\phi)$ in the spherical coordinate system, in free space. Here, $\hat r$ and $\hat\theta$ represent the unit vectors along $r$ and $\theta$, respectively. The value of $P$ in the equation should be ____________ (rounded off to the nearest integer).
$$\bar H(r,\theta,\phi)=\frac1{r^3}\left(\hat rP\cos\theta+\hat\theta\sin\theta\right)$$
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Correct answer: 2
Explanation
In free space $\nabla\cdot\bar H=0$. In spherical coordinates, $\nabla\cdot\bar H=\dfrac1{r^2}\dfrac{\partial}{\partial r}(r^2H_r)+\dfrac1{r\sin\theta}\dfrac{\partial}{\partial\theta}(\sin\theta\,H_\theta)$. With $H_r=P\cos\theta/r^3$ the first term is $-\dfrac{P\cos\theta}{r^4}$, and with $H_\theta=\sin\theta/r^3$ the second is $\dfrac{2\cos\theta}{r^4}$. The sum is $\dfrac{(2-P)\cos\theta}{r^4}=0$, so $P=2$.