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Two indentical magnetic dipoles of magnetic moment $1.0 \mathrm{Am}^{2}$ each,placed at a separation of $2 \mathrm{~m}$ with their axes perpendicular to each other. The resultant magnetic field at a point midway between the dipoles is
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The correct answer is:
$\quad \sqrt{5} \times 10^{-7} T$
Since axes are perpendicular so mid point lies on axial line of one magnet and on equitorial line of other magnet
$\therefore \mathrm{B}_{1}=\frac{\mu_{0}}{4 \pi} \frac{2 \mathrm{M}}{\mathrm{d}^{3}}=\frac{10^{-7} \times 2 \times 1}{1^{3}}=2 \times 10^{-7} \mathrm{~T}$ and $\mathrm{B}_{2}=\frac{\mu_{0}}{4 \pi} \frac{\mathrm{M}}{\mathrm{d}^{3}}=\frac{10^{-7} \times 1}{1^{3}}=10^{-7} \mathrm{~T}$
As $\mathrm{B}_{1} \perp \mathrm{B}_{2}$
$\therefore$ Resultant magnetic field
$=\sqrt{\mathrm{B}_{1}^{2}+\mathrm{B}_{2}^{2}}=\sqrt{5} \times 10^{-7} \mathrm{~T}$
$\therefore \mathrm{B}_{1}=\frac{\mu_{0}}{4 \pi} \frac{2 \mathrm{M}}{\mathrm{d}^{3}}=\frac{10^{-7} \times 2 \times 1}{1^{3}}=2 \times 10^{-7} \mathrm{~T}$ and $\mathrm{B}_{2}=\frac{\mu_{0}}{4 \pi} \frac{\mathrm{M}}{\mathrm{d}^{3}}=\frac{10^{-7} \times 1}{1^{3}}=10^{-7} \mathrm{~T}$
As $\mathrm{B}_{1} \perp \mathrm{B}_{2}$
$\therefore$ Resultant magnetic field
$=\sqrt{\mathrm{B}_{1}^{2}+\mathrm{B}_{2}^{2}}=\sqrt{5} \times 10^{-7} \mathrm{~T}$
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