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Obtain an expression for intensity of el...

Obtain an expression for intensity of electric field in end on position, i.e., axial position of an electric dipole.

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AB is a dipole made up of two charges `-q and +q` separated by a distance 2l.
Let `E_(1) and E_(2)` be the electric field strengths at point P due to charges, +q and -q respectively.

So, electric intensity at due to `-q`
`vec(E)_(1)= (1)/(4pi epsi_(0)).(q)/((r-l)^(2))` (From B to P) and electric intensity at P due to +q
`vec(E )_(2)= (1)/(4pi epsi_(0)).(q)/((r+l)^(2))` (From P to A)
Since `E_(1) gt E_(2)`, the resultant electric field due to the electric dipole has a magnitude equal to difference of `E_(1) and E_(2)` and direction is from B to P.
`therefore E= E_(1)- E_(2)`
`=(1)/(4pi epsi_(0)).(q)/((r-l)^(2))- (1)/(4pi epsi_(0)) .(q)/((r+l)^(2))`
`=(q)/(4pi epsi_(0)) [(1)/((r-l)^(2)) -(1)/((r+l)^(2))]`
`E= (q)/(4pi epsi_(0)) [((r +l)^(2) - (r-l)^(2))/((r^(2)-l^(2))^(2))]`
`= (q)/(4pi epsi_(0)) (r^(2) + l^(2) + 2rl - r^(2)- l^(2) + 2rl)/((r^(2)- l^(2))^(2))`
`=(q)/(4pi epsi_(0)) (4rl)/((r^(2)-l^(2))^(2))`
`=(1)/(4pi epsi_(0)) (2(q xx 2l)r)/((r^(2)-l^(2))^(2))`
But, `q xx 2l= p` (dipole moment)
`therefore E= (1)/(4pi epsi_(0)) (2pr)/((r^(2)-l^(2))^(2))`
If `r gt gt l`, then `E= (1)/(4pi epsi_(0)) (2pr)/(r^(4))`
or `E= (1)/(4pi epsi_(0))(2p)/(r^(3))` directed from B to P
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