Question #5be02

1 Answer
Apr 8, 2017

If color(red) (r <= R)rR
V_r - V_R = Lambda/ (2pivarepsilon_0) [1-r^2/R^2]

If color(red) (r >= R)
V_r - V_R = Lambda/ (2pivarepsilon_0R^2) [ln(R/r)]

Explanation:

We are going to use Gauss's Law: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html

If color(red) (r <= R)

V_r - V_R = int_r^R E_(r) dr

(V_r - V_R ) is the Electrical potential http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elepe.html

E_(r) = (Lambdar)/ (2pivarepsilon_0R^2)

int_r^R (Lambdar)/ (2pivarepsilon_0R^2) dr

Lambda/ (2pivarepsilon_0R^2) int_r^Rrdr

Lambda/ (2pivarepsilon_0R^2) [r^2/2]_r^R

Lambda/ (2pivarepsilon_0R^2) [r^2/2]_r^R

Lambda/ (2pivarepsilon_0R^2) [R^2-r^2]

V_r - V_R = Lambda/ (2pivarepsilon_0) [1-r^2/R^2]

or

If color(red) (r >= R)

E_(r) = Lambda/ (2pivarepsilon_0r)

V_r - V_R = int_r^R E_(r) dr

int_r^R Lambda/ (2pivarepsilon_0r)

Lambda/ (2pivarepsilon_0R^2)int_r^R (1/r)

Lambda/ (2pivarepsilon_0R^2) [ln(r)_r^R]

V_r - V_R = Lambda/ (2pivarepsilon_0R^2) [ln(R/r)]