• # 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 ${{M}_{2}}.$

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]