Question

A long cylinder carries a charge density that is proportional to the distance from the axis r = ks, for some constant k. Find the electric field inside this cylinder.

Answer 1

Answer:

The Electric field inside cylinder is $\dfrac{1}{3}$ k s²    in s^ direction .

Explanation:

Given as :

For a long cylinder

The distance from axis = s

The charge density proportional to distance

i.e  r = k s

where k is constant

Let the electric field inside cylinder = E

According to question

From the Gaussian theorem

Total charge enclosed = $Q_i_n$ = l $\int_{0}^{s}(ks')s'ds'd\phi$

Or,   $Q_i_n$ = l k $\int_{0}^{s}(ks'^{2})ds'd\phi$

Or, $Q_i_n$ = l k $\dfrac{2}{3}$ π s²

i.e $Q_i_n$ =  $\dfrac{2}{3}$ k π s² l

So, The charge enclosed inside cylinder =   $Q_i_n$ =  $\dfrac{2}{3}$ k π s² l   column

Again

∵ Electric field inside cylinder = $\dfrac{Q_in}{\varepsilon _02\Pi sl}$

i.e E = $\dfrac{\frac{2}{3}k\Pi ls^{3}}{\varepsilon _02\Pi sl}$

∴   E = $\dfrac{1}{3}$ k s²    in s^ direction

So, The Electric field inside cylinder =   E = $\dfrac{1}{3}$ k s²    in s^ direction

Hence , The Electric field inside cylinder is $\dfrac{1}{3}$ k s²    in s^ direction . Answer