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
Concept. P is on the axis of the left magnet and on the equatorial line of the right one, $5$ cm from each. Axial field $\dfrac{\mu_0}{4\pi}\dfrac{2m}{r^3}$ points along $\vec m$; equatorial field $\dfrac{\mu_0}{4\pi}\dfrac{m}{r^3}$ points opposite to $\vec m$. The two are perpendicular, so add them as vectors.
Step 1. Axial field. $B_1=10^{-7}\times\dfrac{2\times3\sqrt5}{(0.05)^3}=10^{-7}\times6\sqrt5\times8000=4.8\sqrt5\times10^{-3}$ T, to the right.
Step 2. Equatorial field. Half as large: $B_2=2.4\sqrt5\times10^{-3}$ T, upward (the right magnet's moment points down, S to N).
Step 3. Resultant. $B=\sqrt{B_1^2+B_2^2}=2.4\sqrt5\times10^{-3}\sqrt{4+1}=2.4\times5\times10^{-3}=12\times10^{-3}$ T.
Answer. $\alpha=12$.
Solution by Sreeraj P, M.Sc Physics
