GATE 2020 EE – Question 43
The static electric field inside a dielectric medium with relative permittivity, $\varepsilon_r=2.25$, expressed in cylindrical coordinate system is given by the following expression
$$\mathbf E=\mathbf a_r\,2r+\mathbf a_\varphi\left(\frac3r\right)+\mathbf a_z\,6$$
where $\mathbf a_r,\mathbf a_\varphi,\mathbf a_z$ are unit vectors along r, $\varphi$ and z directions, respectively. If the above expression represents a valid electrostatic field inside the medium, then the volume charge density associated with this field in terms of free space permittivity, $\varepsilon_0$, in SI units is given by:
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Correct answer: (D) $9\varepsilon_0$
Explanation
$\nabla\cdot\mathbf E=\frac1r\frac{\partial(r\cdot2r)}{\partial r}+\frac1r\frac{\partial E_\varphi}{\partial\varphi}+\frac{\partial E_z}{\partial z}=4+0+0=4$. Gauss's law gives $\rho=\varepsilon_r\varepsilon_0\nabla\cdot\mathbf E=2.25\times4\varepsilon_0=9\varepsilon_0$.