Question

Two identical magnetic dipoles of magnetic moments $1.0A-m^2$ each, placed at a separation of 2m with their axis perpendicular to each other. The resultant magnetic field at a point midway between the dipoles is

• A. $5\times {10}^{-7}T$
  
• B. $\sqrt{5}\times {10}^{-7}T$
  
• C. ${10}^{-7}T$
• D. None of these

Answer 1

The correct option is B $\sqrt{5}\times {10}^{-7}T$


With respect to $1^s$t 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.\\
$\therefore B_2=\frac{{\mu }_0}{4\pi }\left(\frac{M}{d^3}\right)\left(Upward\right)$
$B_1={10}^{-7}\times \frac{2\times 1}{1}=2\times {10}^{-7}T,B_2=\frac{B_1}{2}={10}^{-7}T$
As $B_{1}and{B}_2$ are mutually perpendicular, hence the resultant magnetic field
$B_R=\sqrt{B_1^2+B_2^2}=\sqrt{(2\times {10}^{-7}{)}^2+({10}^{-7}{)}^2}=\sqrt{5}\times {10}^{-7}T$