Find electric potential at origin when all negative charges magnitude +Q are placed at 1, 1/3, 1/5, 1/7.. And -Q placed at 1/2, 1/4, 1/6.. Along x-axis.

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Solution

potential , V = KQ/r
$P o t e n t i a l space comma space V equals fraction numerator K Q over denominator r end fraction space space space space space space comma w h e r e space K equals fraction numerator 1 over denominator 4 Ï�Îµ end fraction p o t e n t i a l space d u e space t o space plus Q space a t space 1 comma space 1 divided by 3 space comma space 1 divided by 5 comma space e t c comma V subscript 1 equals fraction numerator K Q over denominator 1 end fraction space plus fraction numerator K Q over denominator 1 divided by 3 end fraction plus fraction numerator K Q over denominator 1 divided by 5 end fraction plus..... space space space equals space K Q space left parenthesis space 1 plus 3 plus 5 plus... space right parenthesis p o t e n t i a l space d u e space t o space minus Q space a t space 1 divided by 2 space comma space 1 divided by 4 comma space 1 divided by 6 space e t c V subscript 2 space equals space fraction numerator negative K Q over denominator 1 divided by 2 end fraction plus fraction numerator negative K Q over denominator 1 divided by 4 end fraction plus fraction numerator negative K Q over denominator 1 divided by 6 end fraction plus... space space space equals space minus K Q space left parenthesis space 2 plus 4 plus 6 plus... space right parenthesis t o t a l space p o t e n t i a l space a t space o r i g i n comma V equals space V subscript 1 plus V 2 space space space equals space space K Q space left parenthesis space 1 plus 3 plus 5 plus... space right parenthesis space space space plus space space space space minus K Q space left parenthesis space 2 plus 4 plus 6 plus... space right parenthesis space space space space equals space space K Q space left parenthesis space 1 plus 3 plus 5 plus... space right parenthesis space space space space plus space space K Q space left parenthesis space minus 2 minus 4 minus 6 minus.... space right parenthesis space space space equals space K Q space space left parenthesis space 1 plus 3 plus 5 plus.... space space minus 2 minus 4 minus 6 minus..... space right parenthesis space space space space equals K Q space left square bracket space space space left parenthesis 1 minus 2 right parenthesis plus left parenthesis 3 minus 4 right parenthesis plus left parenthesis 5 minus 6 right parenthesis plus..... space space right square bracket space space space equals K Q space left parenthesis negative 1 minus 1 minus 1 minus 1 minus negative... right parenthesis space space space equals K Q left parenthesis negative infinity right parenthesis t h e r e space f o r comma space i n i f i t e space n e g a t i v e space p o t e n t i a l space a t space o r i g i n$

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