Two identical short magnetic dipoles of magnetic moment \( 2\,\mathrm{A\,m^{2}} \) are placed with their axis mutually perpendicular and centers 2 m apart. The magnitude of the resultant magnetic field at the midpoint (O) between the dipoles is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( 2\sqrt{5}\times10^{-7}\,\mathrm{T} \)
O is 1 m from the centre of each dipole. For a short dipole the field on the equatorial line is \( B = \mu_0 M/4\pi r^{3} \), directed opposite to the moment, and on the axial line it is twice as large, \( B = \mu_0 \cdot 2M/4\pi r^{3} \), along the moment.
The left dipole has its axis vertical, and O lies on its equatorial line, so its field at O is vertical:
\( \displaystyle B_1 = \frac{\mu_0}{4\pi}\frac{M}{r^{3}} = 10^{-7} \times \frac{2}{1^{3}} = 2\times10^{-7}\,\mathrm{T} \)
The right dipole has its axis horizontal along the line to O, so O lies on its axial line and its field at O is horizontal:
\( \displaystyle B_2 = \frac{\mu_0}{4\pi}\frac{2M}{r^{3}} = 10^{-7} \times \frac{2 \times 2}{1^{3}} = 4\times10^{-7}\,\mathrm{T} \)
The two fields are at right angles, so the resultant is
\( \displaystyle B = \sqrt{B_1^{2} + B_2^{2}} = \sqrt{(2\times10^{-7})^{2} + (4\times10^{-7})^{2}} = \sqrt{20}\times10^{-7} = 2\sqrt{5}\times10^{-7}\,\mathrm{T} \)