Derive an expression for the electric potential at a point along the axial line of an electric dipole.

Explanation

Magnetic dipole: A pair of magnetic poles having equal and opposite pole strength separated by a small distance is known as magnetic dipole. Magnetic field intensity: Magnetic field intensity at a point is defined as the force experienced by a unit north pole when placed at that point.
Magnetic field at a point on the axial line of the magnetic dipole: Let we have a bar magnet NS having length 2l.
O is the mid-point of the magnetic dipole. Let m= Pole strength of each pole of the magnet (magnetic dipole) Let P is any point on the axial line of magnet at a distance r from the centre of magnetic dipole.

Let `OP= r, therefore NP= (r-l) and SP= (r+l)`
Let a unit north pole is placed at P. then magnetic field at P due to N-pole of magnet
`vec(B)_(1)= (mu_(0))/(4pi) (m)/((NP)^(2))= (mu_(0))/(4pi) (m)/((r-l)^(2))` along `vec(NP)`
or `vec(B)_(1)= (mu_(0))/(4pi) (m)/((r-l)^(2))` along `vec(NP)` (produced)
Magnetic field at P due to S-pole of the magnet
`vec(B)_(2)= (mu_(0))/(4pi) (m)/((SP)^(2))` along `vec(PS)`
or `vec(B)_(2)= (mu_(0))/(4pi) (m)/((r+l)^(2))` along `vec(PS)`
As `|vec(B)_(1)| gt |vec(B)_(2)|`
`therefore` Magnetic field at P due to bar magnet
`B= B_(1)- B_(2)` [In magnitude]
`= (mu_(0))/(4pi) [(m)/((r-l)^(2))- (m)/((r+l)^(2))]`
`=(mu_(0)m)/(4pi) [((r+l)^(2)-(r-l)^(2))/((r^(2)-l^(2))^(2))]`
`B= (mu_(0)m)/(4pi) (4rl)/((r^(2)-l^(2))^(2))= (mu_(0))/(4pi) (2r(m.2l))/(4pi(r^(2)-l^(2))^(2)) [because m(2l)= M]`
`therefore B= (mu_(0))/(4pi) (2Mr)/((r^(2)-l^(2))^(2))` along NP produced
For a short magnet or a small magnetic dipole: i.e., when `l^(2) lt lt r^(2)`
Then, `vec(B)= (mu_(0))/(4pi) (2vec(M))/((r^(2))^(2))`
`vec(B)= (mu_(0))/(4pi) (2 vec(Mr))/((r^(2))^(2))`
`vec(B)= (mu_(0))/(4pi) (2vec(M))/(r^(3))`