Question

Electric field due to infinite long straight wire of linear charge density λ varies with ⊥ distance of point from the wire as *

(a) r⁰

(b) r⁻¹

(c) r²

(d) r⁻³​

Answer 1

Given:

A long infinite wire with charge density $\lambda$.

To find:

Relation of field intensity with radial distance r ?

Calculation:

• Consider a cylindrical GAUSSIAN SURFACE around the infinite wire of radius r and height h.

Applying Gauss' Law:

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

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

$\implies \displaystyle \oint E \times ds = \frac{ q_{e} }{ \epsilon_{0}}$

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

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

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

$\implies \displaystyle E \propto {r}^{ - 1}$

So, final answer is :

$\boxed{ \bf E \propto {r}^{ - 1} }$