Question

Derive an expression for electric field intensity at a point on the equitorial line of an electric dipole

Answer 1

Consider an electric dipole consisting of two point changes −q and +q separated by a small distance AB=2a with center O and dipole moment P=q(2a)

Let OP=r

E1= electric intensity at P due to change -q at A.

∴|E1−→|=14π∈0qAP2

AP2=OP2+OA2

=r2+a2

∴|E1−→|=14π∈0qr2+a2

E1−→=Pc−→

Let ∠PBA=∠PAB=θ.

E1 has two components E1 can θ along PR∥BA and E1sinθ along PE⊥BA .

If E2 is the electric intensity at P, due to change +q at B, then

E2−→=14π∈0qBp2=14π∈0q(r2+a2)

E2−→ is along PD−→−

This has two components, E2 is along PR∥BA and E2sinθ along PF.

E→=2E1cosθ

=24π∈0.q(r2+q2)cosθ

=24π∈0.q(r2+a2)(OAAD)

=24π∈0.q(r2+q2)ar2+a2−−−−−√

=q×2a4π∈0(r2+a2)3/2

But q×2a=|P→|, the dipole moment

∴|E→|=|P→|4π∈0(r2+a2)3/2

The direction of E→ is along .

PR−→−||BA−→− (ie) opposite to P→

or E→=−P→4π∈0(r2+a2)3/2