Four identical point charges q are placed at four corners of a square of side a. Find the potential energy of the charge system.
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Explanation

Method 1 (using direct formula) :
`U = U_(12) + U_(13) + U_(14) + U_(23) + U_(24) + U_(34)`
`= (Kq^(2))/(a) + (Kq^(2))/(asqrt(2)) + (Kq^(2))/(a) + (Kq^(2))/(a) + (Kq^(2))/(asqrt(2)) + (Kq^(2))/(a)`
`= [(4Kq^(2))/(a) + (2Kq^(2))/(asqrt(2))] = (2Kq^(2))/(a)[2+(1)/(sqrt(2))]`
Method 2 [using `U = (1)/(2)(U_(1) + U_(2) +...)`] :
`U_(1)` = total P.E. of charge at corner 1 due to all other charges
`U_(2)` = total P.E. of charge at corner 2 due to all other charges
`U_(3)` = total P.E. of charge at corner 3 due to all other charges.
`U_(4)` = total P.E. of charge at corner 4 due to all other charges
Since due to symmetry `U_(1) = U_(2) = U_(3) = U_(4)`
`U_("Net") = (U_(1) + U_(2) + U_(3) + U_(4))/(2) = 2U_(1)`
`=2[(Kq^(2))/(a) + (Kq^(2))/(a) + (Kq^(2))/(sqrt(2)a)] = (2Kq^(2))/(a)[2+(1)/(sqrt(2))]`