The point charges are placed on the vertices of regular hexagon of side `L` as shown. Find the potential energy of the system of charges.

Explanation

Number of pairs `= (6(6-1))/(2) = 15`
(a) Six pairs `(Q , -Q)` at separation `L`.
(b)
Six pairs `(Q , Q : Q , Q: Q , Q , -Q , -Q , -Q , -Q , -Q, -Q)` at separation `sqrt(3) L`.
(c)
Three pairs `(Q , -Q)` at separation `2L`.
`u = (1)/( 4 pi in_(0)) [ (Q(-Q))/(L) xx 6 + (Q.Q)/(sqrt(3) L) xx 6 + ((Q) (-Q))/(2L) xx 3 ]`
`= (Q^(2))/(4 pi in_(0) L) [ - 6 + 2 sqrt(3) - (3)/(2)]`