question_answer

The electric field of an electric dipole at a point on its axis at a distance 'd? from the centre of the dipole varies as

A)  \[1/d\]

B)  \[1/{{d}^{2}}\]

C)  \[1/{{d}^{3}}\]  

D)  \[1/{{d}^{3/2}}\]

Correct Answer: C

Solution :

Electric field of the electric dipole at a point \[P\,(r,\theta )\]is given by \[E=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{p}{{{r}^{3}}}\sqrt{3\,{{\cos }^{2}}\theta +1}\] \[=\frac{1}{4\pi {{\varepsilon }_{0}}}\,\frac{p}{{{d}^{3}}}\,\sqrt{3\,co{{s}^{2}}\,\theta +1}\] \[[\because \,r=d]\] where p is electric dipole moment. On the axis of the dipole \[\theta ={{0}^{o}}\] \[\Rightarrow \] \[E=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{p}{{{d}^{3}}}\times \sqrt{3+1}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{2p}{{{d}^{3}}}\] \[\Rightarrow \] \[E\alpha \frac{1}{{{d}^{3}}}\]