$A$ coil is placed in $y-z$ plane making an angle of $30^{\circ}$ with the $x$-axis. The current through the coil is $I$,and the number of turns is $N$. If a magnetic field of strength $B$ is applied in the positive $x$-direction,find the torque experienced by the coil (Radius of the coil is $R$).
$(N=100, I=1 \text{ A}, R=2 \text{ m}, B=\frac{1}{\pi} \text{ T})$

$A$ rectangular conducting loop consists of two wires on two opposite sides of length $l$ joined together by rods of length $d$. The wires are each of the same material but with cross-sections differing by a factor of $2$. The thicker wire has a resistance $R$ and the rods are of low resistance,which in turn are connected to a constant voltage source $V_{0}$. The loop is placed in a uniform magnetic field $\vec{B}$ at $45^{\circ}$ to its plane. Find the torque exerted by the magnetic field on the loop about an axis through the centers of the rods.

$A$ closely wound solenoid of $800$ turns and area of cross section $2.5 \times 10^{-4} \;m^{2}$ carries a current of $3.0 \;A$. If the solenoid is free to turn about the vertical direction and a uniform horizontal magnetic field of $0.25 \;T$ is applied,what is the magnitude of torque on the solenoid when its axis makes an angle of $30^{\circ}$ with the direction of the applied field?

$A$ circular coil of $20$ turns and radius $10\; cm$ is placed in a uniform magnetic field of $0.10\; T$ normal to the plane of the coil. If the current in the coil is $5.0\; A$,what is the
$(a)$ total torque on the coil,
$(b)$ total force on the coil,
$(c)$ average force on each electron in the coil due to the magnetic field?
(The coil is made of copper wire of cross-sectional area $10^{-5} \;m ^{2},$ and the free electron density in copper is given to be about $10^{29}\; m ^{-3}.$)

$A$ galvanometer coil has $500$ turns and each turn has an average area of $3 \times 10^{-4} \ m^{2}$. If a torque of $1.5 \ Nm$ is required to keep this coil parallel to the magnetic field when a current of $0.5 \ A$ is flowing through it,the strength of the magnetic field (in $T$) is:

$A$ coil in the shape of an equilateral triangle of side $2 \,cm$ is suspended from a vertex such that it hangs in a vertical plane between the poles of a permanent magnet producing a horizontal magnetic field of $100 \times 10^{-3} \,T$. The magnetic field is parallel to the plane of the coil. For the moment of couple acting on the coil to be $2 \sqrt{3} \times 10^{-5} \,Nm$, the current to be passed through the coil is (in $A$)