Question

Two magnets of moment $M$ and $2\;M$ are tied at an angle of ${60^0}$ to each other. What is the resultant magnetic moment of the combination?
(A) $\sqrt 7 M$
(B) $\sqrt 8 M$
(C) $\sqrt 9 M$
(D) $\sqrt {10} M$

Answer 1

Hint: The quantity that defines the ability of the magnetic force to exert a force on electric currents and the torque that the magnetic will exert on it. The direction of the magnetic moment is from the south pole to the north pole of the magnet. Here we have two magnets and we have to find the resultant magnetic moment of the two magnets.

Formula used:
Total magnetic moment
${M_T} = \sqrt {M_1^2 + M_2^2 + 2{M_1}{M_2}\cos \theta } $
where ${M_T}$ stands for the total magnetic moment, ${M_1}$ and ${M_2}$ stands for the magnetic moments of the two magnets, and $\theta $ stands for the angle between the two angles.

Complete Step by step solution:
The magnetic moment of the two magnets is given as $M$ and $2\;M$. The two magnets are said to be tied together.
The angle between the two magnets is given as ${60^0}$.
The resultant magnetic moment of the two magnets at an angle ${60^0}$ can be written as,
${M_R} = \sqrt {{M^2} + {{\left( {2M} \right)}^2} + 2 \times (M \times 2M)\cos {{60}^0}} $
We know that $\cos {60^0} = \dfrac{1}{2}$
Substituting this in the above equation, we get
${M_R} = \sqrt {{M^2} + 4{M^2} + 4{M^2} \times \dfrac{1}{2}} $
This will become,
${M_R} = \sqrt {{M^2} + 4{M^2} + 2{M^2}} = \sqrt 7 M$
Therefore, the answer is: Option (A): $\sqrt 7 M$.

Additional information
The limits of a current loop or the poles of the magnets are defined by the magnetic moment. A magnet placed in a magnetic field will be experiencing a torque. When the dipole is perpendicular to the magnetic field, the potential energy of the dipole is assumed to be zero. The magnetic moment will give the orientation and magnetic strength of the magnet. Devices called magnetometers are used to measure magnetic moments.

Note:
It is the magnetic moment that actually denotes the magnetic dipole moment of the magnet. The magnetic field of the magnet will be directly proportional to the dipole moment of the magnet. Dipole moment actually refers to the ability of any object to align with the magnetic field.