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Two identical magnetic dipoles of magnetic moments $1.0 \,A-m^2$ each, placed at a separation of $2\,m$ with their axis perpendicular to each other. The resultant magnetic field at a point midway between the dipoles is
$5 \times {10^{ - 7}}\,T$
$\sqrt 5 \times {10^{ - 7}}\,T$
${10^{ - 7}}\,T$
None of these
Solution
(b)With respect to $1^{st} $ magnet, $P$ lies in end side-on position
$\therefore {B_1} = \frac{{{\mu _0}}}{{4\pi }}\left( {\frac{{2M}}{{{d^3}}}} \right)$ $(RHS)$
With respect to $2^{nd} $ magnet. $P$ lies in broad side on position.
$\therefore \;{B_2} = \frac{{{\mu _0}}}{{4\pi }}\left( {\frac{M}{{{d^3}}}} \right)$ (Upward)
${B_1} = {10^{ – 7}} \times \frac{{2 \times 1}}{1} = 2 \times {10^{ – 7}}T,\;{B_2} = \frac{{{B_1}}}{2} = {10^{ – 7}}\,T$
As $B_1$ and $B_2$ are mutually perpendicular, hence the resultant magnetic field
${B_R} = \sqrt {B_1^2 + B_2^2} = \sqrt {{{(2 \times {{10}^{ – 7}})}^2} + {{({{10}^{ – 7}})}^2}} $$ = \sqrt 5 \times {10^{ – 7}}\,T$
