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Two charges -q and +q are located at poi...

Two charges -q and +q are located at points (0,0,-a) and (0,0,a) respectively.
(b) Obtain the dependence of potential on the distance r of a point from the origin when `r gt gt a`.

Text Solution

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Let P be the point at a distance r from the dipole. OP makes an angle `theta` with the axis of the dipole. The length of the dipole = 2a.
`r_(2) = BP ~~ CP = r - a cos theta`
`r_(1) AP ~~ DP = r + a cos theta`
Potential at P,
`V = V_(1) + V_(2) = q/(4pi epsilon_(0))(1/(r_2)-1/(r_1))`
`=q/(4 pi epsilon_(0))(1/(r-acostheta)-1/(r+a cos theta))`

`:. V = q/(4 pi epsilon_(0)) (2 a cos theta)/(r^(2)-a^(2)cos^(2)theta)`
`=(2 aq cos theta)/(4 pi epsilon_(0)) {1/(r^2)(1-(a^2)/(r^2) cos^(2)theta)^(-1)}`
`~~(p cos theta)/(4 pi epsilon_(0)r^(2))[ :' (a^2)/(r^2) lt lt 1]`
`:. V prop 1/(r^2)`.
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