Electric Field at a point on the equatorial line

The perpendicular bisector of the line joining the charges is the equatorial line. Q is a point at distance r from the centre O of the dipole on the equatorial line.



Let E1 and E2 denote the electric fields due to charges + q and –q respectively at the point Q. The magnitudes of E1 and E2 are equal.

E1 = E2 = ____1____         ___q___
        _______4 π ε0             (d2 + r2 )

Now we resolve E1 and E2 into components along the X and Y directions. The Y components of E1 and E2 cancel out. But the X components of E1 and E2 add to yield the resultant field E. Thus

    E = E1 cos Ф + E2 cos Ф = 2 E1 cos Ф         (E1 = E2)

But cos Ф = _____d_____
           ______ (d2 + r2)1/2
    E = 2. ____1____ __q___ .  ____d____ = _____q(2d)_______
           ___4 π ε₀       (d² + r²)        (d² + r²)^1/2   4 π ε₀  (d² + r²)^1/2
      =  _____1_____ p
        _____4 π ε₀     r³
Direction of E is opposite to that of dipole moment vector p.


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