What is the force between two magnetic dipoles?

A coil of $50$ turns carrying a current of $2\mathrm{A}$ in a magnetic field of $0.5\mathrm{T}$. The torque acting on the coil is

A $100$ turns coil shown in figure carries a current of $2 \mathrm{A}$ in a magnetic field $B=0.2 \mathrm{Wb} {\mathrm{m}}^{-2}.$ The torque acting on the coil is

The rectangular coil of area $A$ is in a field $B.$ Find the torque about the $Z$-axis when the coil lies in the position shown and carries a current $I.$

A magnetic dipole is oscillating in a magnetic field obeying the following expression

$\frac{{\mathrm{d}}^2\mathrm{\theta }}{{\mathrm{dt}}^2}=-\frac{\mathrm{mB}}{\mathrm{I}}\mathrm{\theta }$:

What is the time period of oscillation and mention the nature of oscillation?

A bar magnet is oscillating in Earth's magnetic field with the time period $\mathrm{T}$. If a similar magnet with the same mass and dimensions has magnetic dipole moment $4$ times that of this magnet, then the periodic time will be______.

Two short magnetic dipoles $m_1$ and $m_2$ each having magnetic moment of $1 \mathrm{A} {\mathrm{m}}^2$ are placed at point $O$ and $P$ respectively. The distance between $OP$ is $1 \mathrm{m}$. The torque experienced by the magnetic dipole $m_2$ due to the presence of $m_1$ is $\_\_\_\_\_\_\_\_\times {10}^{-7}\mathrm{N} \mathrm{m}$.

An electric dipole of moment $\overrightarrow{\mathrm{P}}$ is placed in a uniformed magnetic field $\overrightarrow{\mathrm{E}}$. The dipole is rotated through a very small angle $\theta$ from equilibrium and is released. Prove that it executes simple harmonic motion with periodic time 

$\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{I}}{\mathrm{PE}}}$ where $\mathrm{I}$= moment of inertia of the dipole.

The rectangular loop of dimensions $0.4 \mathrm{m}\times 0.2 \mathrm{m}$ shown in the figure consists of $50$ closely wrapped turns and is hinged along the $z$-axis. The plane of the loop makes an angle of $30°$ with the $y$-axis. The current in the windings is $0.5 \mathrm{A}$.

The torque exerted on the loop in this position by a magnetic field $\overrightarrow{B}=\hat{j}1.0 \mathrm{T}$

A square loop of dimensions $0.4 \mathrm{m}\times 0.3 \mathrm{m}$ consists of $100$ closely wrapped turns. The loop is hinged along the $y$-axis. A uniform magnetic field of $0.75 \mathrm{T}$ is directed along the $x$-axis.

How much current should flow in the loop so that the maximum torque exerted on the loop is $18 \mathrm{N}-\mathrm{m}?$