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 symmetery, `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)]`