question_answer

Figure shows an electric quadrupole, with quadruple moment \[(Q\text{ }=\text{ }2q{{\ell }^{2}}).\]The electric field at a distance from its center at the axis of the quadrupole is given by

A) \[\left( \frac{1}{4\pi {{\in }_{0}}} \right)\frac{Q}{{{r}^{4}}}\]

B) \[\left( \frac{1}{4\pi {{\in }_{0}}} \right)\frac{2Q}{{{r}^{4}}}\]

C) \[\left( \frac{1}{4\pi {{\in }_{0}}} \right)\frac{3Q}{{{r}^{4}}}\] 

D) None of these  

Correct Answer: C

Solution :

[c] \[E=\frac{1}{4\pi {{\in }_{0}}}\left[ \frac{2p}{{{\left( r-\ell /2 \right)}^{3}}}-\frac{2p}{{{\left( r+\ell /2 \right)}^{3}}} \right]\] \[=\frac{1}{4\pi {{\in }_{0}}}\frac{3Q}{{{r}^{4}}}.\]