Question

Question #5be02

Answer 1

If `color(red) (r <= R)`
`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)]`