A current-carrying conductor is in the form of a sine curve as shown, which carries current $I$. If the equation of this curve is $y=2\sin \left(\frac{\pi x}{L}\right)$ and a uniform magnetic field exists in space, then,

          

The figure shows a rectangular $20$-turn coil of wire of dimensions $10 \mathrm{cm}$ by $5.0 \mathrm{cm}$. It carries a current of $0.10 \mathrm{A}$ and is hinged along one long side. It is mounted in the $xy$ plane at an angle $\theta =30°$ to the direction of a uniform magnetic field of magnitude $0.50 \mathrm{T}$. In unit vector notation, what is the torque acting on the coil about the hinge line?

A current loop is carrying a current of $5.0 A$, which is in the shape of a right angled triangle with sides $30, 40$ and $50 \mathrm{cm}$. The loop is in a uniform magnetic field of magnitude $80 \mathrm{mT}$ whose direction is in parallel to the current in the $50 \mathrm{cm}$ side of the loop. Find the magnitude of the magnetic dipole moment of the loop.

A current loop, carrying a current of $5.0 \mathrm{A}$, is in the shape of a right-angled triangle with sides $30 \mathrm{cm}, 40 \mathrm{cm}$ and $50 \mathrm{cm}$. The loop is in a uniform magnetic field of magnitude $80 \mathrm{mT}$ whose direction is parallel to the current in the $50 \mathrm{cm}$ side of the loop. Find the magnitude of the torque acting on the loop.

The coil in figure carries current $i=2.00 A$ in the direction indicated, is parallel to an $xz$ plane, has $3.00$ turns and an area of $4.00\times {10}^{-3}{\mathrm{m}}^2$,, and lies in a uniform magnetic field $\overrightarrow{\mathrm{B}}=(2.00\mathrm{f}-3.00\mathrm{j}-4.00\hat{\mathrm{k}})\mathrm{mT}$. What are

$\left(a\right)$ the orientation energy of the coil in the magnetic field and

$\left(b\right)$ the torque (in unit-vector notation) on the coil due to the magnetic field?

The coil in the figure carries a current $i=2.00 \mathrm{A}$ in the direction indicated, is parallel to an $xz$ plane, has $3.00$ turns and an area of $4.00\times {10}^{-3} {\mathrm{m}}^2$, and lies in a uniform magnetic field $\overrightarrow{B}\mathit{=}(2.00\hat{\mathrm{i}}-3.00\hat{\mathrm{j}}-4.00\hat{\mathrm{k}}) \mathrm{mT}$. What is the torque (in unit-vector notation) on the coil due to the magnetic field?

In figure (a), it has two concentric coils, lying in the same plane, carry current in opposite directions. The current in the larger coil $1$ is fixed. The current $i_2$ in coil $2$ can be varied. In figure (b) it gives the net magnetic moment of the two-coil system as a function of $i_2$. The vertical axis of the scale is set by ${\mu }_{net,S}=2.0\times {10}^{-5} \mathrm{A} {\mathrm{m}}^2$, and the horizontal axis scale is set by $i_{2S}=10.0 \mathrm{mA}$. If the current in coil $2$ is then reversed, what is the magnitude of the net magnetic moment of the two-coil system when $i_2=7.0 \mathrm{mA}$?

The figure gives the net magnetic moment of the two-coil system as a function of $i_2$. The vertical axis scale is set by ${\mu }_{net}=2.0\times {10}^{-5} \mathrm{A} {\mathrm{m}}^2$ and the horizontal scale is set by $i_{2_s}=10.0 \mathrm{m} \mathrm{A}$. If the current in coil $2$ is reversed, then what is the magnitude of the net magnetic moment of the two coil system when $i_2=7.0 \mathrm{m} \mathrm{A}$?

A spherical shell of radius $\text{'}R\text{'}$ and uniformly charged with charge $\text{'}Q\text{'}$ is rotating about its axis with frequency $\text{'}f\text{'}$. Find the magnetic moment of the sphere.