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Question

# The electric potential due to an electric dipole at a distance r from the centre of the dipole and at an angle θ to the dipole axis is (p is the dipole moment, 2a is the length of the dipole) :

A
V=pcosθ4πϵo(r2+a2cos2θ)
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B
V=pcosθ4πϵo(r2a2cos2θ)
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C
V=pcosθ4πϵo(r2a2sin2θ)
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D
V=pcosθ4πϵo(r2+a2sin2θ)
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Solution

## The correct option is B V=pcosθ4πϵo(r2−a2cos2θ)From figure, cosθ=r2+a2−r2+2raWe get r+=(r2+a2−2racosθ)1/2As r>>a, we can neglect a2 as compared to r2∴ r+=(r2−2racosθ)1/2Or r+=r(1−2arcosθ)1/2Using (1+x)n=1+nx for x<<1We get r+=r(1−arcosθ)⟹ r+=r−acosθSimilarly, r−=r+acosθElectric potential at point A V=Vq+V−q∴ V=q4πϵor+−q4πϵor−Or V=q4πϵo[1r+−1r−]Or V=q4πϵo[1r−acosθ−1r+acosθ]Or V=q4πϵo[2acosθ(r−acosθ)(r+acosθ)]⟹ V=pcosθ4πϵo(r2−a2cos2θ)

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