Question

drive an expression for electric field of a dipole on equatorial plane

Answer 1

Consider an electric dipole consisting of
two point changes − q and + q separated
by a small distance AB =2 a with center
O and dipole moment P =q (2 a )
Let OP =r
E 1= electric intensity at P due to change
-q at A.
∴|
→
E 1 |= 1
4 π ∈ 0
q
AP 2
AP 2= OP 2+ OA 2
= r 2 + a2
∴|
→
E 1 |= 1
4 π ∈ 0
q
r 2+ a 2
→
E 1=
→
Pc
Let ∠PBA =∠PAB = θ.
E 1 has two components E 1 can θ along
PR ∥ BA and E 1 sinθ along PE ⊥BA .
If E 2 is the electric intensity at P, due to
change +q at B, then
→
E 2= 1
4 π ∈0
q
Bp 2 = 1
4 π ∈0
q
(r 2+ a 2)
→
E 2 is along
→
PD
This has two components, E 2 is along
PR ∥ BA and E 2 sinθ along PF.
→
E =2 E1 cos θ
= 2
4π ∈0
. q
( r 2 + q2 )
cos θ
= 2
4π ∈0
. q
( r 2 + a2 ) (OA
AD )
= 2
4π ∈0
. q
( r 2 + q2 )
a
√ r 2 +a 2
= q ×2 a
4π ∈0 (r 2 +a 2 )3/2
But q ×2 a =|
→
P |, the dipole moment
∴|
→
E |= |
→
P |
4π ∈0 (r 2 +a 2 )3/2
The direction of
→
E is along .
→
PR ||
→
BA (ie) opposite to
→
P
or
→
E = −
→
P
4 π ∈0 (r 2+ a 2) 3/2
If the dipole is short 2 a<< r
|
→
E |= |
→
P |
4 π ∈0 r 3
|
→
E | α 1
r 3
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