GATE 2022 EE – Question 45
If the magnetic field intensity ($\mathbf H$) in a conducting region is given by the expression, $\mathbf H=x^2\hat\imath+x^2y^2\hat\jmath+x^2y^2z^2\hat k$ A/m. The magnitude of the current density, in A/m$^2$, at $x=1$ m, $y=2$ m, and $z=1$ m, is
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Correct answer: (B) 12
Explanation
$\mathbf J=\nabla\times\mathbf H$ has components $J_x=\frac{\partial H_z}{\partial y}-\frac{\partial H_y}{\partial z}=2x^2yz^2=4$, $J_y=\frac{\partial H_x}{\partial z}-\frac{\partial H_z}{\partial x}=-2xy^2z^2=-8$ and $J_z=\frac{\partial H_y}{\partial x}-\frac{\partial H_x}{\partial y}=2xy^2=8$ at $(1,2,1)$. The magnitude is $\sqrt{16+64+64}=12$ A/m$^2$.