A steady current $\mathrm{I}$ flows along an infinitely long hollow cylindrical conductor of radius $\mathrm{R}$. This cylinder is placed coaxially inside an infinite solenoid of radius $2\mathrm{R}$. The solenoid has $\mathrm{n}$ turns per unit length and carries a steady current $\mathrm{I}$. Consider a point $\mathrm{P}$ at a distance r from the common axis. The correct statement(s) is (are) :

The figure shows a circular loop of radius a with two long parallel wires (numbered $1$ and $2$) all in the plane of the paper. The distance of each wire from the centre of the loop is $d$. The loop and the wires are carrying the same current $I$. The current in the loop is in the counterclockwise direction if seen from above.

Consider $d>>a$, and the loop is rotated about its diameter parallel to the wires by $30°$ from the position shown in the figure. If the currents in the wires are in the opposite directions, the torque on the loop at its new position will be (assume that the net field due to the wires is constant over the loop)

A conductor (shown in the figure) carrying constant current $I$ is kept in the $x-y$plane in a uniform magnetic field $\overrightarrow{B}$ . If $F$ is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is (are)

A charged particle (electron or proton) is introduced at the origin $\left(x=0,y=0,z=0\right)$ with a given initial velocity. $\overrightarrow{v}$. A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$, electric field $\overrightarrow{E}$ and magnetic field $\overrightarrow{B}$ are given in column $1,2$ and $3$, respectively. The quantities $E_0,B_0$ are positive in magnitude.

In which case will the particle move in a straight line with constant velocity ?

A charged particle (electron or proton) is introduced at the origin $\left(x=0,y=0,z=0\right)$ with a given initial velocity. $\overrightarrow{v}$. A uniform electric field $\overrightarrow{E}$ and a uniform magnetic field $\overrightarrow{B}$ exist everywhere. The velocity $\overrightarrow{v}$, electric field $\overrightarrow{E}$ and magnetic field $\overrightarrow{B}$ are given in column $1,2$ and $3$, respectively. The quantities $E_0,B_0$ are positive in magnitude.

In which case would the particle move in a straight line along the negative direction of $y-$axis, (i.e, move along $– \hat{y}$)? 

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $4a$. The magnitude of the magnetic field at the centre of the loop is

A uniform magnetic field $B$ exists in the region between $x=0$ and $x=\frac{3R}{2}$ (region $2$ in the figure) pointing normally into the plane of the paper. A particle with charge $+Q$and momentum $p$ directed along $\mathrm{x}$-axis enters region $2$ from region $1$ at point $P_1 (y = –R).$ Which of the following option(s) is/are correct ?

In the $xy$-plane, the region $y>0$ has a uniform magnetic field $B_1\widehat{ \mathrm{k}}$ and the region $y<0$ has another uniform magnetic field $B_2 \widehat{ \mathrm{k}}$. A positively charged particle is projected from the origin along the positive $y-$axis with speed $v_0=\pi {\mathrm{ms}}^{-1} \text{at} t=0$, as shown in the figure. Neglect gravity in this problem. Let $t=T$ be the time when the particle crosses the$x-$axis from below for the first time. If $B_2=4B_1$, the average speed of the particle, in ${\mathrm{ms}}^{-1}$, along the$x-$axis in the time interval $T$ is __________.