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Question
Answer the following question.
Derive an expression for the electric field at any point on the equatorial line of an electric dipole.
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Solution
Electric field for points on the equator plane
The magnitude of the electric field due to two charges +q and –q is
`E_(+q) = q/(4piε_0)1/(r^2 + a^2)`
`E_(-q) = q/(4piε_0)1/(r^2 + a^2)`
`E_(+q) = E_(-q)`
The directions of `E_(+q) and E_(-q)`, are as shown in the figure. The components normal to the dipole axis cancel away. The components along the dipole axis add up.
Total electric field, `E = -(E_(+q) + E_(-q)) costheta hatp` ......(Negative sign shows that the field is opposite to `hatp`)
`E = -(2qa)/(4piε_0(r^2 + a^2)^(3/2))hatp` ...(iii)
At large distance (r >> a), this reduces to,
`E = -(2qa)/(4piε_0r^3)hatp`
`vecp = q xx 2vecahatp`
`E = -vecp/(4piε_0r^3)` ....(r >> a)
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