Answer to Question #175404 in Physics for Rishi

Question #175404

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



1
Expert's answer
2021-03-26T11:39:00-0400

Write down Gauss's theorem as he wrote it:


(dPdx+dQdy+dRdz)dω=(Pcosα+Qcosβ+Rcosγ)ds\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:


EdS=Qnetϵ0.\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


EdS=EdScos0°, EdS=EdS=E(4πR2), \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


EdS=Qnetϵ0, E(4πR2)=Qnetϵ0, E=14πϵ0QR2.\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}.


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