Question

Write down gauses theorem and stablish coulombs inverse square law by using this theorem

Answer 1

Write down Gauss's theorem as he wrote it:

$$\int\bigg(\frac{dP}{dx}+\frac{dQ}{dy}+\frac{dR}{dz}\bigg)d\omega=\int(P\cos\alpha+Q\cos\beta+R\cos\gamma)ds$$

However, another formula is famous in the field of electromagnetism:

$$\oint\vec{E}\vec{dS}=\frac{Q_\text{net}}{\epsilon_0}.$$

Obtain inverse square law. For the electric field from a point charge, we have

$$\oint\vec{E}\vec{dS}=\int EdS\cos0°,\\\space\\ \oint\vec{E}\vec{dS}=E\int dS=E(4\pi R^2),\\\space\\$$

with Gauss's theorem we have

$$\oint\vec{E}\vec{dS}=\frac{Q_\text{net}}{\epsilon_0},\\\space\\ E(4\pi R^2)=\frac{Q_\text{net}}{\epsilon_0},\\\space\\ E=\frac{1}{4\pi\epsilon_0}\frac{Q}{R^2}.$$