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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.
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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