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### The electric field at an equatorial point of a dipole:

As shown in fig consider an electric dipole consisting of charges -q and +q, separated by distance 2a and placed in a vacuum. Let P be a point on the equatorial line of the dipole at distance r from it.

i.e. OP=r

The electric field at an equatorial point of a dipole |

By geometry ∠PAO = ∠RPA (∴ PR ॥ AO)

and ∠PBO = ∠ MPR

Simulation of Electric field at an Equatorial point of a dipole |

The electric field at point P due to +q charge is

E+q

→

=14Ï€Ïµ0.qr2+a2)−−−−−−(√

2=14Ï€Ïµ0.q ,direction along BPr2+a2

→

=14Ï€Ïµ0.qr2+a2)−−−−−−(√

2=14Ï€Ïµ0.q ,direction along BPr2+a2

The electric field at point P due to -Q charge is

E−q

→

=14Ï€Ïµ0.qr2+a2)−−−−−−(√

2=14Ï€Ïµ0.q ,direction along PAr2+a2

→

=14Ï€Ïµ0.qr2+a2)−−−−−−(√

2=14Ï€Ïµ0.q ,direction along PAr2+a2

Thus the magnitudes of E+q and E-q are equal i.e.

E E+q−→

Clearly, the normal components of E-q and E+q will cancel out. The horizontal component or parallel components to the dipole axis add up. The total electric field Eequa−→−− is opposite to p⃗

Eequa−→−−=−(E−qcosÎ¸+E+qcosÎ¸).p ^

=−2E−qcosÎ¸p^

Eequa−→−−=−14Ï€Ïµ0.p(a2+r2)3/2p^

Where p=2qa, is the electric dipole moment.

If the point p is located far away from the dipole,

r>>a, then

Eequa−→−−=−14Ï€Ïµ0.pr3p^

Clearly, the direction of the electric field at any point on the equatorial line of the dipole will be antiparallel to the dipole moment p⃗

### Comparison of an electric field of dipole at axial and equatorial points.

Eaxial=2Eequatorial

the electric field of a short dipole at a distance r along its axis is twice the electric field at the same distance along the equatorial line.