Gauss' Law

Electric Field induced by a Point Charge

In electromagnetic theory, the electric field created by a point charge $q$ located at the origin is

E(x, y, z) = \frac{1}{4\pi \epsilon\_0} \frac{q}{|r|^2} \left(\frac{r}{|r|}\right) = \frac{q}{4\pi \epsilon\_0}\frac{r}{|r|^3} = \frac{q}{4\pi \epsilon\_0}\frac{xi + yj + zk}{\rho^3} \= \frac{q}{4\pi \epsilon\_0}\vec{F}

• $ϵ\_0$ physical constant
• $r$ position vector of point mass
• $\rho$ being the radius of the sphere

Analyzing the Electric Field using the Divergence Theorem

As the outward flux of a field $\vec{F}=\frac{xi+yj+zk}{{\rho }^3}$ across any sphere centered at the origin is $4\pi$, the outward flux of the field $\frac{q}{4\pi ϵ\_0}\vec{F}$ is $\frac{\rho }{ϵ\_0}$. Because the divergence $\nabla \cdot \vec{E}=\nabla \cdot \frac{q}{4\pi ϵ\_0}\vec{F}=0$ when $\rho >0$ then $\iint \_\text{Boundary of D}E\cdot nd\sigma =0$, which tells that the flux of $E$ across $S\_1$ in the direction away from the origin must be the same as the flux of E across $S\_2$ ($S\_1$, an inner smaller sphere and $S\_2$, an outer bigger sphere enclose a 3d-space in between them, which’s 2d-boundary is $S$) - (For the evaluation of flux at least a two dimensional surface is required.)

Gauss’ Law

Gauss Law builds upon the principles discovered in Analyzing the Electric Field using the Divergence Theorem and states that the outward flux for any closed surface that encloses the origin

$$\iint \_S\vec{E}\cdot \vec{n}d\sigma =\frac{q}{ϵ\_0}$$

holds.

Requirements

• $S$ encloses the origin