Derivation of electric potential energy of a system of two point charges in the absence of external electric field

_{1 , }q

_{2}

_{,}located at r

_{1}and r

_{2}respectively. The work done in bringing charge q

_{1}from infinity to r

_{1 }is zero because there is no potential in absence of electrical field.

Work done in bring the charge q

_{2}from infinty to r

_{2}against the field due to charge q

_{1}

_{ }is = q

_{1}V

where V is potential at point r

_{2}due to the charge q

_{1}.

As, V= q

_{1}/4$\pi \epsilon $

_{0}r

_{12}

where 'r

_{12}' is the distance between charge 'q

_{1}' and 'q

_{2}'.

Therefore potential energy of the system is given by the total work done in bring the two charges from infinity to their respective points.

U = 0 + q

_{2}q

_{1}/4$\pi \epsilon $

_{0}r

_{12}

U = q

_{2}q

_{1}/4$\pi \epsilon $

_{0}r

_{12}

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