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Derive the expression for the electric ...

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

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Electric potential due to a dipole at a point on the end-on position: Figure below shows a dipole consisting of two equal and opposite charges +q and -q separated by a distance 2l. Let P be a point on the end-on position (on the axis) of the dipole. OP is the axial line of the dipole.

The potential due to the charge +q at the point `P= V_(1)`
`V_(1)= (1)/(4pi epsi_(0)) (q)/(r-1)`
The potential due to the charge -q at the point `P= V_(2)`
`V_(2)= (1)/(4pi epsi_(0)).((-q))/((r+l))`
Total potential at the point P `V= V_(1) + V_(2)`
`=(1)/(4pi epsi_(0)) (q)/(r-l) + (1)/(4pi epsi_(0)) ((-q))/((r+l))`
`=(q)/(4pi epsi_(0)) [(1)/(r-l) -(1)/(r + l)]`
`=(q)/(4pi epsi_(0)) [(r+l - r+ l)/((r -l) (r +l))]`
`= (1)/(4pi epsi_(0)).(2lq)/(r^(2)-l^(2))`
But the dipole moment p= 2lq
So, `V= (1)/(4pi epsi_(0)) .(p)/((r^(2)-l^(2)))`
Suppose `r gt gt l`. Then `l^(2)` can be neglected compared to `r^(2)`. So for a short dipole, the electric potential at a point on the end-on position is,
`V= (1)/(4pi epsi_(0)) .(p)/(r^(2))`
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