Three identical dipoles with charges q and `-q`, and separation A between the charges, are placed at the corners of an equilateral triangle of side D as shown in fig. Find the interaction energy of the system `(a lt lt d)`.
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Answer

`(6q^2 a^2)/(4 pi epsilon_0 d^3)`

Explanation

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Electric field `E_1` due to dipole at `B` is
`E_1 = (k 2 p cos 60^(@) )/(d^3)` and `E_2 = (k sin 60^(@))/(d^3)`
Potential energy between `A and B` is
`- vec p . vec E = - vec p.(vec E_1 + vec E_2) = - vec p. vec E_1` , `(because vec p _|_ vec E_2)`
or `-p E_1 cos 180^(@) = p E_1 = (k 2 p^2 cos 60^(@))/(d^3) = (q^2 a^2)/(4 pi epsilon_0 d^3)`
Net energy of system is `3 q^2 a^2 // 4 pi epsilon_0 d^3`.