use gauss's law Will the electric field of a cylinder that is charged non-uniformly, namely with a charge density p = Po • r² , behavedifferently with distance than a uniformly charged cylinder?1. What about inside?2. What about outside?

Answered: Will the electric field of a cylinder…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

use gauss's law

Will the electric field of a cylinder that is charged non-uniformly, namely with a charge density p = Po • r² , behave
differently with distance than a uniformly charged cylinder?
1. What about inside?
2. What about outside?

Expert Solution

Step 1

Electric field due to non-uniformly charged cylinder

Given that,

charge density $ρ = ρ_{0} . r^{2}$

Consider a point p inside the cylinder with radius vector r. To calculate E at P, we take the Gaussian surface to be a shell of radius r_g.

The non-uniform of height h and radius r.

The area enclosed by the cylinder is

$A = 2 πrh$

Charge enclosed by the surface is

$Q_{en} = ρ . V If dr is the thickness of the shell and h is the height then, Substituting the given values in the equation we get {dQ}_{en} = ρ_{0} r^{2} . 2 πhr . dr Q_{en} = \int_{0}^{r_{g}} ρ_{0} 2 {πhr}^{3} . dr = ρ_{0} 2 πh \frac{r^{4}}{4} = \frac{{πr}^{4} {hρ}_{0}}{2}$