Gauss's Law

All the charge given to a conducting sphere resides on outer surface.

$\text{?}\mathrm{}\stackrel{?}{\mathrm{E}}\cdot \mathrm{d}\stackrel{?}{\mathrm{s}}=0\Rightarrow {?}_{\text{net}}={?}_{\text{in}}-{?}_{\text{out}}=0$

$\Rightarrow {?}_{\text{in}}={?}_{\text{out}}$

Using gauss law in differential form

$\overrightarrow{\nabla }.\overrightarrow{E}=\frac{\rho }{{\epsilon }_0}$

$\overrightarrow{\nabla }.\left({\epsilon }_0\overrightarrow{E}\right)=\rho$

$\Rightarrow \overrightarrow{\nabla }.\overrightarrow{D}=\rho$

$\Rightarrow \left(\frac{\partial }{\partial x}\widehat{i}+\frac{\partial }{\partial y}\widehat{j}+\frac{\partial }{\partial z}\widehat{k}\right).\overrightarrow{D}=\rho$

$\begin{array}{l}\Rightarrow \rho =-e^{-x}siny+e^{-x}siny+2=2\\ \Rightarrow Q=2\left(Volume\right)=4*10^{-9}c=4nc\end{array}$

Direction of E in the direction of y axes so flux is only due to top and bottom surface for bottom surface y = 0 E = 0

and for top surface y = 0.5 m      So

$E=150*{\left(0.5\right)}^2=\frac{150}{4}$

$fluxflowing?=EA=\frac{150}{4}*{\left(0.5\right)}^2$              

$=\frac{150}{16}$             

Gausses law $?=\frac{q}{{\epsilon }_0}$  

  $\frac{150}{16}=\frac{q}{{\epsilon }_0}$            

$=8.3*10^{-11}C$              

Gauss law for magnetism states that total magnetic flux through a closed surface S will be zero. It is one of the four maxwell's equations that is useful in understanding the prinicples of electromagnetism. When understood in depth, the law implies that magnetic field lines form closed loops since they are continous. In short, it confirms that magnetic monopoles cannot exist.