Question

The electric field created by a line charge falls with distance r from line charge as (a) 1/r (b) 1/r^2 (c) r (d) r^0​

Answer 1

Given:

A line charge is provided.

To find:

Electric field varies with distance as?

Calculation:

• Let's assume a cylindrical GAUSSIAN SURFACE around the line charge with uniform linear charge density $\lambda$.

Applying Gauss' Law:

$\displaystyle \oint \vec{E}. \vec{ds} = \frac{ q_{e} }{ \epsilon_{0}}$

$\implies \displaystyle \oint E \times ds \times \cos( {0}^{ \circ} ) = \frac{ \lambda h}{ \epsilon_{0}}$

$\implies \displaystyle \oint E \times ds = \frac{ \lambda h}{ \epsilon_{0}}$

$\implies \displaystyle E \oint ds = \frac{ \lambda h}{ \epsilon_{0}}$

$\implies \displaystyle E \times 2\pi rh = \frac{ \lambda h}{ \epsilon_{0}}$

$\implies \displaystyle E = \frac{ \lambda }{ 2\pi r\epsilon_{0}}$

$\implies \displaystyle E \propto \frac{1 }{ r}$

So, E is proportional to 1/r.