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

column $I$	column $II$	column $III$
$(I)$ Electron with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$	$(i)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0^2 \hat{\mathrm{Z}}$	$(P)$ $\overrightarrow{\mathrm{B}}=-\mathrm{B}_0 \hat{\mathrm{x}}$
$(II)$ Electron with $\overrightarrow{\mathrm{v}}=\frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{y}}$	$(ii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{y}}$	$(Q)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{x}}$
$(III)$ Proton with $\overrightarrow{\mathrm{v}}=0$	$(iii)$ $\overrightarrow{\mathrm{E}}=-\mathrm{E}_0 \hat{\mathrm{x}}$	$(R)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{y}}$
$(IV)$ Proton with $\overrightarrow{\mathrm{v}}=2 \frac{\mathrm{E}_0}{\mathrm{~B}_0} \hat{\mathrm{x}}$	$(iv)$ $\overrightarrow{\mathrm{E}}=\mathrm{E}_0 \hat{\mathrm{x}}$	$(S)$ $\overrightarrow{\mathrm{B}}=\mathrm{B}_0 \hat{\mathrm{z}}$

($1$) In which case will the particle move in a straight line with constant velocity?

$[A] (II) (iii) (S)$    $[B] (IV) (i) (S)$   $[C] (III) (ii) (R)$   $[D] (III) (iii) (P)$

($2$) In which case will the particle describe a helical path with axis along the positive $z$ direction?

$[A] (II) (ii) (R)$   $[B] (IV) (ii) (R)$  $[C] (IV) (i) (S)$   $[D] (III) (iii)(P)$

($3$)  In which case would be particle move in a straight line along the negative direction of y-axis (i.e., more along $-\hat{y}$ )?

$[A] (IV) (ii) (S)$   $[B] (III) (ii) (P)$   $[C]$ (II) (iii) $(Q)$   $[D] (III) (ii) (R)$

• A

  $A,C,D$

• B

  $A,C$

• C

  $C,D$

• D

  $B,C$