Question #5be02

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Answer 1 · 2017-04-08T23:26:00

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

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

If $color(red) (r <= R)$

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

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

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

$Lambda/ (2pivarepsilon_0R^2) int_r^Rrdr$

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

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

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

or

If $color(red) (r >= R)$

$E_(r) = Lambda/ (2pivarepsilon_0r)$

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

$int_r^R Lambda/ (2pivarepsilon_0r)$

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

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

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