Two identical small bar magnets each of dipole moment $3\sqrt{5}$ J/T are placed at a center to center separation of $10$ cm, with their axes perpendicular to each other as shown in figure. The value of magnetic field at the point P midway between the magnets is $\alpha \times 10^{-3}$ T. The value of $\alpha$ is ______
$(\mu_0 = 4\pi \times 10^{-7}$ Tm/A$)$
Numerical value type. Enter your answer.
Answer: 12
$P$ is $r = 5$ cm from each magnet, and $\dfrac{\mu_0}{4\pi}\cdot\dfrac{M}{r^3} = 10^{-7} \times \dfrac{3\sqrt{5}}{1.25 \times 10^{-4}} = 2.4\sqrt{5} \times 10^{-3}$ T.
$P$ lies on the axis of the left magnet: $B_1 = 2 \times 2.4\sqrt{5} \times 10^{-3} = 4.8\sqrt{5} \times 10^{-3}$ T (horizontal).
$P$ lies on the equatorial line of the right magnet: $B_2 = 2.4\sqrt{5} \times 10^{-3}$ T (vertical).
The two are perpendicular: $B = \sqrt{5} \times 10^{-3}\sqrt{4.8^2 + 2.4^2} = \sqrt{5} \times 2.4\sqrt{5} \times 10^{-3} = 12 \times 10^{-3}$ T.
Solution by Sreeraj P, M.Sc Physics