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.

	Column-1		Column-2		Column-3
(I)	Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(i)	$\overrightarrow{E}=E_0\widehat{z}$	(P)	$\overrightarrow{B}=-B_0\widehat{x}$
(II)	Electron with $\overrightarrow{v}=\frac{E_0}{B_0}\widehat{y}$	(ii)	$\overrightarrow{E}=-E_0\widehat{y}$	(Q)	$\overrightarrow{B}=B_0\widehat{x}$
(III)	Electron with $\overrightarrow{v}=0$	(iii)	$\overrightarrow{E}=-E_0\widehat{x}$	(R)	$\overrightarrow{B}=B_0\widehat{y}$
(IV)	Electron with $\overrightarrow{v}=2\frac{E_0}{B_0}\widehat{x}$	(iv)	$\overrightarrow{E}=E_0\widehat{x}$	(S)	$\overrightarrow{B}=B_0\widehat{z}$

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 __________.