GATE 2022 EC – Question 65
In an electrostatic field, the electric displacement density vector, $\vec D$, is given by
$$\vec D(x,y,z)=(x^3\,\vec\imath+y^3\,\vec\jmath+xy^2\,\vec k)\ \text{C/m}^2,$$
where $\vec\imath,\vec\jmath,\vec k$ are the unit vectors along $x$-axis, $y$-axis, and $z$-axis, respectively. Consider a cubical region $R$ centered at the origin with each side of length 1 m, and vertices at $(\pm0.5\text{ m},\pm0.5\text{ m},\pm0.5\text{ m})$. The electric charge enclosed within $R$ is ________ C (rounded off to two decimal places).
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Correct answer: 0.49 to 0.51
Explanation
By Gauss's law the enclosed charge is $\iiint\nabla\cdot\vec D\,dV$ with $\nabla\cdot\vec D=3x^2+3y^2+0$. Over the unit cube centred at the origin $\iiint x^2\,dV=\int_{-0.5}^{0.5}x^2dx=\frac1{12}$, and the same for $y^2$. So $Q=3\cdot\frac1{12}+3\cdot\frac1{12}=0.5$ C.