question_answer

Two circular loops, of radii r and 2r, have currents I and I/2 flowing through them in clockwise and anti-clockwise sense, respectively. If their equivalent magnetic moments are \[{{M}_{1}}\] and \[{{M}_{2}}\]respectively, then state the relation between \[{{M}_{1}}\] and \[{{M}_{2}}.\]

Answer:

Radius of circular loops are \[{{r}_{1}}=r\]and \[{{r}_{2}}=2r;{{I}_{1}}=I\]and \[{{I}_{2}}=\frac{I}{2}\]             Magnetic moment of the loops are                         \[{{M}_{1}}={{\mu }_{0}}\times {{I}_{1}}\times \pi r_{1}^{2}={{\mu }_{0}}I\pi {{r}^{2}}\]                         \[{{M}_{2}}={{\mu }_{0}}\frac{I}{2}\pi {{(2r)}^{2}}=2{{\mu }_{0}}I\pi {{r}^{2}}\]             \[\therefore \]      \[\frac{{{M}_{1}}}{{{M}_{2}}}=\frac{{{\mu }_{0}}I\pi {{r}^{2}}}{2{{\mu }_{0}}I\pi {{r}^{2}}}=\frac{1}{2}\Rightarrow {{M}_{2}}=2{{M}_{1}}\]             Also,     \[{{M}_{2}}=-\,2{{M}_{1}}\] [\[\because \] Currents \[{{I}_{1}}\] and \[{{I}_{2}}\] are opposite in direction]