Torque on a Current Loop

A circular loop of area $1 {\mathrm{cm}}^2,$ carrying a current of $10 \mathrm{A}$ is placed in a magnetic field of $0.1 \mathrm{T}$, perpendicular to the plane of the loop. The torque on the loop due to the magnetic field is

The torque on a current loop is zero when the angle between the positive normal and the magnetic field is either $\mathrm{\theta }=0$ or $\mathrm{\theta }=180°$. In which of these two orientations, the equilibrium is stable?

A rectangular coil of $100$ turns has length $5\mathrm{cm}$ and width $4\mathrm{cm}.$ It is placed with its plane parallel to a uniform magnetic field and a current of $2 \mathrm{A}$ is sent through the coil. Find the magnitude of the magnetic field $\mathrm{B}$, if the torque acting on the coil is $0.2 \mathrm{N} \mathrm{m}$.

A $50$ -turn circular coil of radius $2.0$ cm carrying a current of $5.0 \mathrm{A}$ is rotated in a magnetic field of strength $0.20 \mathrm{T}.$

(a) What is the maximum torque that acts on the coil?

(b) In a particular position of the coil, the torque acting on it is half of this maximum. What is the angle between the magnetic field and the plane of the coil?

A rectangular loop of sides $20\mathrm{cm}$ and $10\mathrm{cm}$ carries a current of $5.0 \mathrm{A}$. A uniform magnetic field of magnitude $0.20 \mathrm{T}$ exists parallel to the longer side of the loop.
(a) What is the force acting on the loop?
(b) What is the torque acting on the loop?

A circular coil of radius $2.0 \mathrm{cm}$ has $500$ turns in it and carries a current of $1.0 \mathrm{A}$. Its axis makes an angle of $30°$with the uniform magnetic field of magnitude $0.40 \mathrm{T}$ that exists in the space. Find the torque acting on the coil.

A circular loop carrying current $i$ has wire of total length $L$. A uniform magnetic field $B$ exists parallel to the plane of the loop.

(a) Find the torque on the loop.
(b) If the same length of the wire is used to form a square loop, what would be the torque? Which is larger?

A square coil of edge $\mathrm{I}$ having $\mathrm{n}$ turns carries a current $\mathrm{i}$. It is kept on a smooth horizontal plate. A uniform magnetic field $\mathrm{B}$ exists in a direction parallel to an edge. The total mass of the coil is $\mathrm{M}$. What should be the minimum value of $\mathrm{B}$ for which the coil will start tipping over?

Consider a nonconducting ring of radius $\mathrm{r}$ and mass $\mathrm{m}$ which has a total charge $\mathrm{q}$ distributed uniformly on it. The ring is rotated about its axis with an angular speed, $\omega$
(a) Find the equivalent electric current in the ring.
(b) Find the magnetic moment $\mu$ of the ring.
(c) Show that $\mu =\frac{\mathrm{q}}{2\mathrm{m}}l$ where $I$ is the angular momentum of the ring about its axis of rotation.

Consider a nonconducting plate of radius $r$ and mass $\mathrm{m}$ which has a charge $\mathrm{q}$ distributed uniformly over it. The plate is rotated about its axis with an angular speed $\omega$. Show that the magnetic moment $\mu$ and the angular momentum $I$ of the plate are related as $\mu =\frac{q}{2m}l$.

Consider a solid sphere of radius r and mass m which has a charge q distributed uniformly over its volume. The sphere is rotated about a diameter with an angular speed $\omega$, Show that the magnetic moment $\mu$ and the angular momentum $I$ of the sphere are related as $\mu =\frac{q}{2m}l$.