[Physics] Electric Field at surface/side of cylinder

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I know I can use Gauss's law to find the Electric Field inside and outside the cylinder very easily. We can select Gaussian surfaces for different cases (i.e. $r \lt R$ and $r \gt R$, where $R$ is the radius of the cylinder and $r$ is the radius of Gaussian surface).

But I want to know Electric fields at the bottom and the upper circular surface of the cylinder.

Edit: Charge of the cylinder can be anything ($\lambda, \sigma, \rho$). For clarification there is one cylinder and I want to find the E field on the top and bottom circular surfaces rather than inside or outside.

Best Answer

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$K=\frac1{4\pi\epsilon_0}$

Using elemental rings in disc we can get the Electric field at point P.

$dE = \frac{Kdq}{r^2}$ where $dq = \sigma.2\pi xdx $

Integrating from 0 to R we get

$E = \frac{Kqr}{(R^2+r^2)^{3/2}}$ where $q = \sigma\pi R^2$ $ = $

Using this in the cylinder

$dE = \frac{Kqx}{(x^2+R^2)^{3/2}}$ where $q = \rho\pi R^2dx$

Integrating this field from 0 to $\ell$ gives the required result.

$E = \frac{\rho R^2}{4\epsilon_0}\{\frac 1R - \frac 1{(R^2+l^2)^{3/2}}\}$ $ = \frac{\rho R}{4\epsilon_0}\{1 - \frac 1{R(1+\frac{l^2}{R^2})^{3/2}}\} $

and for $l\to\infty$

$E = \frac{\rho R}{4\epsilon_0}$