question_answer

What is the mutual inductance of a two-loop system as shown with centre separation l

A)             \[\frac{{{\mu }_{0}}\pi {{a}^{4}}}{8{{l}^{3}}}\]

B)             \[\frac{{{\mu }_{0}}\pi {{a}^{4}}}{4{{l}^{3}}}\]

C)             \[\frac{{{\mu }_{0}}\pi {{a}^{4}}}{6{{l}^{3}}}\]

D)             \[\frac{{{\mu }_{0}}\pi {{a}^{4}}}{2{{l}^{3}}}\]

Correct Answer: D

Solution :

                   Magnetic field at the location of coil (2) produced due to coil (1) \[{{B}_{1}}=\frac{{{\mu }_{o}}}{4\pi }.\frac{2M}{{{l}^{3}}}\] Flux linked with coil (2) \[{{\varphi }_{2}}={{B}_{1}}{{A}_{2}}=\frac{{{\mu }_{o}}}{4\pi }\frac{2i\ (\pi {{a}^{2}})}{{{l}^{3}}}\times (\pi {{a}^{2}})\] Also \[{{\varphi }_{2}}=Mi\] \[\Rightarrow M=\frac{{{\mu }_{o}}\pi {{a}^{4}}}{2{{l}^{3}}}\]