Question

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

Answer 1

Answer: 4

Electric flux density
$(D)=\frac{\text{ charge }}{\text{ Area }}\times \hat{r}=\frac{Q}{4\pi r^2}\hat{r}={ϵ}_0\left(\frac{Q}{4\pi {ϵ}_0{r}^2}\hat{r}\right)$
$\Rightarrow \vec{E}=\frac{\vec{D}}{{ϵ}_0}=\frac{e^{-x}\sin y\hat{i}-e^{-x}\cos y\hat{j}+2z\hat{k}}{{\in }_0}$
Also by Gauss's law
$\frac{\rho }{{ϵ}_0}=\left(\frac{\partial }{\partial x}\hat{i}+\frac{\partial }{\partial y}\hat{j}+\frac{\partial }{\partial z}\hat{k}\right)\cdot \vec{E}=\left(\frac{\partial }{\partial x}\hat{i}+\frac{\partial }{\partial y}\hat{j}+\frac{\partial }{\partial z}\hat{k}\right)\cdot \frac{\vec{D}}{{ϵ}_0}$
$\Rightarrow \rho =\frac{\partial }{\partial x}\left(e^{-x}\sin y\right)+\frac{\partial }{\partial y}\left(-e^{-x}\cos y\right)+\frac{\partial }{\partial z}(2z)$
$\rho =-e^{-x}\sin y+e^{-x}\sin y+2$
At origin $\rho =-e^{-0}\sin 0+e^{-0}\sin 0+2$
$\rho =2C/m^3$
Charge $=\rho \times$ volume
$=2\times 2\times 10^{-9}=4\times 10^{-9}=4nC$