Question #5be02

1 Answer
Apr 8, 2017

Answer:

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)]#