Question

(a) Define a law of electrostatics in integral form that is used to compute the electrostatic field E due to a symmetric distribution of charge within a given volume. State the meaning of the terms in the defining equation. (b) A uniformly charged long cylinder of radius a and length L has total charge q inside its volume. What is the direction of the electric field at points outside the cylinder? Find the electric field inside and outside the cylinder. (c) Total charge q is distributed along a line of length L. Determine the electric field produced by the wire using Gauss’s law. Compare this result with the result in part (b).

Answer 1

A)

$\oint_S \vec E\vec{dS} = \iiint_V div(\vec E) dV$ - Gauss theorem

$\iiint_V div(\vec E) dV = -4\pi\iiint_V \rho dV$ - one of Maxwell equations

$-\frac{1}{4\pi}\oint_S \vec E\vec{dS} = \iiint_V \rho dV$

This equation says

The flow of the E-field through a surface S increase if and only if increase the total charge of volume V bonded by a surface S.

B) Cylinder (R - a radius, L - a length) case:

$\oint_S \vec E\vec{dS} = 2\pi LrE_r = -\frac{4\pi q}{\pi R^2 L}V_{cyl} = -4\pi q$

There is $E_r$ - a radial component of the E - field

$E_r = -\frac{2q}{rL}$, $r\leq R$

$E_r = -\frac{2q}{RL}, r> R$

C) Line cases(L - a length) case:

$\oint_S \vec E\vec{dS} = 2\pi LrE_r = -4\pi q$

$E_r = -\frac{2q}{rL}$

We assume q<0 in both cases