Gauss's Law

Gauss Law is only concerned with the total enclosed charge that finally tells us the total flux. The charges outside may change field patterns. They not affect the total flux. It's actually incorrect to assume the field due to the external charges should also affect the flux through the Gaussian surface. 

Gauss Law does not directly give the electric field in all cases. It can only be used in calculations for symmetrical surfaces: spherical, cylindrical, or planar. 

The integral form of Gauss Law is considered as an indirect form and only in theory. It will still create a mathematical problem. The Gaussian surface passing through a discrete charge means it lies on the surface. Half of the electric flux is outside and half in. Not on the boundary. And we know Gauss' Law holds true only when there are closed surfaces. 

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.