derive equation of eletric field due to dipole at axis and eqator?
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Explanation:Electric field at A due to the charges q and −q is shown in the above figure. We resolve E into horizontal and vertical components. The vertical components (Esinθ) cancel out each other and only the horizontal components survive to give net electric field at A as 2Ecosθ.
Electric field at point A, E
A
=2Ecosθ
where E=
4πϵ
o
(r
2
+a
2
)
q
We get E
A
=
4πϵ
o
(r
2
+a
2
)
2q
cosθ
From figure, we have cosθ=
r
2
+a
2
a
∴ E
A
=
4πϵ
o
(r
2
+a
2
)
3/2
2qa
We know dipole moment P=2qa
⟹ E
A
=
4πϵ
o
(r
2
+a
2
)
3/2
P
For a<<r, we can neglect a
2
compared to r
2
.
⟹ E
A
=
4πϵ
o
r
3
P
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