Question

The total charge enclosed in an incremental volume of $2\times {10}^{-9}{\text{m}}^3$ located at the origin is$\text{nC}$, if electric flux density of its field is found as $D=e^{-x}\sin \hat{i}-e^{-x}\cos y\hat{j}+2z\hat{k}{\text{C/m}}^2$.

Answer 1

By Gauss's law, volume charge density is given by

$\rho =\Delta .\overrightarrow{E}=(\frac{\partial }{\partial x}\hat{i}+\frac{\partial }{\partial y}\hat{j}+\frac{\partial }{\partial z}\hat{k}).\overrightarrow{D}$

$\Rightarrow \rho =\frac{\partial }{\partial x}\left(e^{-x}\sin y\right)+\frac{\partial }{\partial y}(-e^{-x}\cos y)+\frac{\partial }{\partial z}\left(2z\right)$

$\Rightarrow \rho =-e^{-x}\sin y+e^{-x}\sin y+2$

At origin, $\rho =-e^{-0}\sin 0+e^{-0}\sin 0+2=2{\text{C/m}}^3$

$\therefore$ Charge $=\rho \times$ volume $=2\times 2\times {10}^{-9}=4\times {10}^{-9}=4\text{nC}$

Correct answer: $4$