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

Explanation

Electric field intensity at a point is defined as the force experienced by a unit positive charge placed at that point.
In S.I., the unit of E is (`NC^(-1)` or `Vm^(-1)`).
Axial line of an electric dipole is a line joining the centre of two charges and extended on either side. It is also known as End-on-position.
Consider an electric dipole consisting of charges `- q and +q`, separated by distance 2a and placed in the free space. Let P be a point on the axial line joining the two charges of the dipole at a distance r from the centre O of the dipole.
So `OP = r, AP = r + a` and `BP = r - a `
The electric field `vecE` at point P due to dipole will be the resultant of the electric field `vecE_A` (due to charge - q at point A) and `vecE_B` (due to charge + q at point B) i.e.,
`vecE = vec(E_A) + vec(E_B)`
Now, `|vecE_A| = 1/(4pi in_0) . q/(AP^2) = 1/(4pi in_0) . (q)/((r + a)^2) " " ` (along PA)
and `|vecE_B| = 1/(4pi in_0) . q/(BP^2) = 1/(4pi in_0) . q/((r - a)^2) " " `(along PX)
Obviously `|vecE_B|` is greater than `|vecE_A|` .
Since `vecE_A` and `vecE_B`, act along the same line but in opposite direction, the magnitude of the net electric field at point P is given by

or `E = 1/(4pi in_0) . q/((r - a)^2) - 1/(4pi in_0) . q/((r + a)^2) " " ` (along PX)
`= 1/(4 pi in_0) . q[((r + a)^2 - (r - a)^2)/((r^2 - a^2)^2)]`
or `E = 1/(4 pi in_0) . (q(4ra))/((r^2 - a^2)^2) = 1/(4pi in_0) (q(2a)(2r))/((r^2 - a^2)^2)`
Now, `q (2a)= p`, the magnitude of the electric dipole moment of the dipole. Therefore,
`E = 1/(4pi in_0) . (2pr)/((r^2 - a^2)^2)` (along PX)
It may be noted that direction of electric field at a point on axial line of the dipole is from charge `-q` to `+q`, same as that of electric dipole moment of the dipole. If dipole is of very short length or when the point P is very far away from the electric dipole, then `r^2 gt gt a^2` and hence `a^2` can be neglected
`E = 1/(4pi in_0) . (2pi r)/(r^4)`
`E = 1/(4pi in_0) . (2pi r)/(r^3)`