Question

State Gauss's theorem in electrostatics using
this theorem derire an expression for the electric
field due to a thin infinite sheet of having uniform density?

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Answer 1

                                        Explanation

Gauss's Law

The total electric flux through a closed surface ( 3- D Surface ) is 1/E₀ time the net charge enclosed by the surface

⇒ Electric field due to an Infinite long uniformly charged wire ( at a distance r )

$\Phi_{net}=\Phi_1+\Phi_2+\Phi_3$

$\rm at \theta =90^{o} \\\Phi_1 \: and \: \Phi_2=0$

So

$\phi_3=\oint EdA\cos0^{o}$

$\phi_{net}=\oint EdA\cos0^{o}$

$\phi_{net}=\oint EdA$

$\phi_{net}=E\oint dA$

$\phi_{net}=E\times2\pi rl$

$\Phi_{net\: Gauss \:law}=\dfrac{q_{in}}{E_o} =\dfrac{\lambda l}{E_o}$

$\Phi_{net}=E2\pi rl$

$\dfrac{\lambda l}{E_o} =E2\pi rl$

$\boxed{\rm E=\dfrac{\lambda}{2\pi E_o r } }$