Question

deduce an expression for the electric potential due to an electro dipole at an equatorial plane ?​

Answer 1

Explanation:

Let an electric dipole consist of two equal and opposite point charges –q at A and +q at B ,separated by a small distance AB =2a ,with centre at O.

The dipole moment, p=q×2a

We will calculate potential at any point P, where

OP=r and ∠BOP=θ

Let BP=r

1

and AP=r

2

Draw AC perpendicular PQ and BD perpendicular PO

In ΔAOC cosθ=OC/OA=OC/a

OC=acosθ

Similarly, OD=acosθ

Potential at P due to +q=

4πϵ

0

1

r

2

q

And Potential at P due to −q=

4πϵ

0

1

r

1

q

Net potential at P due to the dipole

V=

4πϵ

0

1

(

r

2

q

−

r

1

q

)

⟹V=

4πϵ

0

q

(

r

2

1

−

r

1

1

)

Now, r

1

=AP=CP

=OP+OC

=r+acosθ

And r

2

=BP=DP

=OP–OD

=r−acosθ

V=

4πϵ

0

q

(

r−acosθ

1

−

r+acosθ

1

)

=

4πϵ

0

q

(

r

2

−a

2

cos

2

θ

2acosθ

)

=

r

2

−a

2

cos

2

θ

pcosθ

(Since p=2aq)