Part A A very long solid cylinder of radius R has positive charge uniformly dist

Question

Part A A very long solid cylinder of radius R has positive charge uniformly distributed throughout it, with charge per unit volume ?. From the expression for E for this cylinder: E(r)=?r2??0R2 for r?R and E(r)=?2??0rfor r?R, find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length ? of the charge distribution. Express your answer in terms of the given quantities and appropriate constants

Graph E as a function from 0 to 3R Graph V as a function from 0 to 3R
Part A A very long solid cylinder of radius R has positive charge uniformly distributed throughout it, with charge per unit volume ?. From the expression for E for this cylinder: E(r)=?r2??0R2 for r?R and E(r)=?2??0rfor r?R, find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length ? of the charge distribution. Express your answer in terms of the given quantities and appropriate constants Part A A very long solid cylinder of radius R has positive charge uniformly distributed throughout it, with charge per unit volume ?. From the expression for E for this cylinder: E(r)=?r2??0R2 for r?R and E(r)=?2??0rfor r?R, find the expressions for the electric potential V as a function of r, both inside and outside the cylinder. Let V = 0 at the surface of the cylinder. In each case, express your result in terms of the charge per unit length ? of the charge distribution. Express your answer in terms of the given quantities and appropriate constants

Graph E as a function from 0 to 3R Graph V as a function from 0 to 3R


Graph E as a function from 0 to 3R Graph V as a function from 0 to 3R

Explanation / Answer

For the first part at a distance r from the center the enclosed charge is ?*p*r^2*L (where L is the length of the Gaussian surface

and A = 2pr*L

So E = (?*p*r^2*L)/(2peo*r*L) = ?*r/(2eo)

In part two the enclosed charge is ?*p*R^2*L and the Gaussian surface is 2pr*L

So E = ?*p*R^2*L/(2peo*r*L) = ?*R^2/(2eo*r)

But in terms of charge per unit length ? (= q/L)

we get ? = ?*p*R^2

So E = ?/(2peo*r)

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