Question

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

Answer 1

Electric flux density
$(D)=\frac{\text { charge }}{\text { Area }} \times \hat{r}=\frac{Q}{4 \pi r^{2}} \hat{r}=\epsilon_{0}\left(\frac{Q}{4 \pi \epsilon_{0} r^{2}} \hat{r}\right) $
$\Rightarrow \vec{E}=\frac{\vec{D}}{\epsilon_{0}}=\frac{e^{-x} \sin y \hat{i}-e^{-x} \cos y \hat{j}+2 z \hat{k}}{\in_{0}}$
Also by Gauss's law
$\frac{\rho}{\epsilon_{0}}=\left(\frac{\partial}{\partial x} \hat{i}+\frac{\partial}{\partial y} \hat{j}+\frac{\partial}{\partial z} \hat{k}\right) \cdot \vec{E}=\left(\frac{\partial}{\partial x} \hat{i}+\frac{\partial}{\partial y} \hat{j}+\frac{\partial}{\partial z} \hat{k}\right) \cdot \frac{\vec{D}}{\epsilon_{0}} $
$\Rightarrow \rho=\frac{\partial}{\partial x}\left(e^{-x} \sin y\right)+\frac{\partial}{\partial y}\left(-e^{-x} \cos y\right)+\frac{\partial}{\partial z}(2 z) $
$\rho=-e^{-x} \sin y+e^{-x} \sin y+2 $
At origin $ \rho=-e^{-0} \sin 0+e^{-0} \sin 0+2$
$\rho=2 C / m^{3} $
Charge $=\rho \times $ volume
$=2 \times 2 \times 10^{-9}=4 \times 10^{-9}=4 \,n C$