question_answer

A thin circular wire carrying a current I has a magnetic moment M. The shape of the wire is changed to a square and it carries the same current. It will have a magnetic moment             [MP PET 2003; MP PMT 2004]

A)            M  

B)            \[\frac{4}{{{\pi }^{2}}}M\]

C)            \[\frac{4}{\pi }M\]             

D)            \[\frac{\pi }{4}M\]

Correct Answer: D

Solution :

                   Initially for circular coil \[L=2\pi r\]and \[M=i\times \pi {{r}^{2}}\] \[=i\times \pi {{\left( \frac{L}{2\pi } \right)}^{2}}=\frac{i{{L}^{2}}}{4\pi }\]  ..... (i) Finally for square coil \[{M}'=i\times {{\left( \frac{L}{4} \right)}^{2}}=\frac{i{{L}^{2}}}{16}\]         ..... (ii) Solving equation (i) and (ii) \[{M}'=\frac{\pi M}{4}\]