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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
→
=14πϵ0.qr2+a2)−−−−−−(√
2=14πϵ0.q ,direction along BPr2+a2
→
=14πϵ0.qr2+a2)−−−−−−(√
2=14πϵ0.q ,direction along BPr2+a2
The electric field at point P due to -Q charge is
E−q
→
=14πϵ0.qr2+a2)−−−−−−(√
2=14πϵ0.q ,direction along PAr2+a2
→
=14πϵ0.qr2+a2)−−−−−−(√
2=14πϵ0.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−qcosθ+E+qcosθ).p ^
=−2E−qcosθp^
Eequa−→−−=−14πϵ0.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−→−−=−14πϵ0.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.