In the figure shown a positively charged particle of charge $q$ and mass $m$ enters into a uniform magnetic field of strength $B$ as shown in the figure. The magnetic field points inwards and is present only within a region of width $d$. The initial velocity of the particle is perpendicular to the magnetic field and $\varphi =30º$. Find the time spent by the particle inside the magnetic field if $d=0.2 \mathrm{m}, B=1 \mathrm{T}, q=1 \mathrm{C}$, $m=1 \mathrm{kg}$ and $v=1 \mathrm{m} {\mathrm{s}}^{-1}$. Use $\sin 45°=0.7$, if required.

A thin beam of charged particles is incident normally on the boundary of a region containing a uniform magnetic field as shown. The beam comprise of mono energetic $\mathrm{\alpha }$ and ${\mathrm{\beta }}^{-}$ particles, each possessing a kinetic energy $\mathrm{T}$. Neglecting interaction between particles of the beam, find the separation between the points on line $\mathrm{PQ}$where the $\mathrm{\alpha }$ and $\mathrm{\beta }$ particles emerge out of the magnetic field. (express your answer in terms of $\mathrm{T}, \mathrm{B},\mathrm{\alpha } (2\mathrm{e}, {\mathrm{m}}_{\mathrm{\alpha }}) \& {\mathrm{\beta }}^{–} (–\mathrm{e}, {\mathrm{m}}_{\mathrm{e}} ),$ where all terms have their usual meanings.)

In the figure shown an infinitely long wire carries a current ${\mathrm{I}}_1$ and another uniform rigid wire $\mathrm{ACB}$ (bent at $‘\mathrm{C}’$ at right angle) of mass $‘\mathrm{m}’$carries a current ${\mathrm{I}}_2$, it is hinged at corner $‘\mathrm{C}’$. Find the angular acceleration of this wire $\mathrm{ACB}$ just after release. Is the direction of rotation clockwise or anticlockwise. (Assume gravity is absent)$[\mathrm{Take} l \mathrm{n}2 = 0.7]$

A loop $\mathrm{PQR}$ formed by three identical uniform conducting rods each of length $\text{'} \mathrm{a} \text{'}$is suspended from one of its vertices $\left(\mathrm{P}\right)$ so that it can rotate about horizontal fixed smooth axis $\mathrm{CD}$. Initially plane of loop is in vertical plane. A constant current $\text{'} \mathrm{i} \text{'}$is flowing in the loop. Total mass of the loop is $\text{'} \mathrm{m} \text{'}$. At$\mathrm{t} = 0$, a uniform magnetic field of strength $\mathrm{B}$ directed vertically upwards is switched on. Acceleration due to gravity is $\text{'} \mathrm{g} \text{'}$. Then find the minimum value of $\mathrm{B}$ so that the plane of the loop becomes horizontal (even for an instant) during its subsequent motion.

A finite conductor $\mathrm{AB}$ carrying current $\mathrm{i}$is placed near a fixed very long wire current carrying ${\mathrm{i}}_0$ as shown in the figure. Find the point of application and magnitude of the net ampere force on the conductor $\mathrm{AB}$. What happens to the conductor $\mathrm{AB}$ if it is free to move. (Neglect gravitational field)

A current of $10 \mathrm{A}$flows around a closed path in a circuit which is in the horizontal plane. The circuit consists of eight alternating arcs of radii ${\mathrm{r}}_1 = 0.08 \mathrm{m}$ and ${\mathrm{r}}_2 = 0.12 \mathrm{m}$. Each arc subtends the same angle at the centre.

(i) Find the magnetic field produced by this circuit at the centre.

(ii) An infinitely long straight wire carrying a current of $10 \mathrm{A}$ is passing through the centre of the given circuit vertically with the direction of the current being into the plane of the circuit. What is the force acting on the wire at the centre due to the current in the circuit? What is the force acting on the arc $\mathrm{AC}$ and the straight segment $\mathrm{CD}$ due to the current in the central wire.