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Question
Deduce the expression for the potential energy of a system of two point charges q1 and q2 brought from infinity to the points `vecr_1`and `vecr_2`respectively in the presence of external electric field `vecE`.
Solution
The work done in bringing charge q1 from infinity to `vecr` is q1V(`vecr`).
Work done on q2 against external field = q2 V(`vecr`)
Work done on q2 against the field due to q1`= (q_1q_1)/(4piepsi_0r_12)`
Where, r12 is the distance between q1 and q2.
By the superposition principle for fields,
Work done in bringing q2 to`vecr_2`is`(q_2V(vecr_2) + (q_1q_2)/(4piepsi_0r_12))`.
Thus,
Potential energy of system = The total work done in assembling the configuration
`=q_1V(vecr_1)+q_2V(vecr_2)+(q_1q_1)/(4piepsi_0r_12)`
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