$A$ particle of charge $q$ and mass $m$ is moving along the $x-$ axis with a velocity $v,$ and enters a region of electric field $E$ and magnetic field $B$ as shown in the figures below. For which figure the net force on the charge may be zero?

The energies required to set up in a cube of side $10 \,cm$ $(i)$ a uniform electric field of $10^7 \,Vm^{-1}$ and (ii) a uniform magnetic field of $0.25 \,Wbm^{-2}$ are respectively about $(\mu_0=4 \pi \times 10^{-7} \,Hm^{-1}, \varepsilon_0=8.9 \times 10^{-12} \,Fm^{-1})$

$A$ proton moving with a constant velocity passes through a region of space without any change in its velocity. If $\overrightarrow{E}$ and $\overrightarrow{B}$ represent the electric and magnetic fields respectively,then this region of space may have

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

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

$(1)$ In which case will the particle move in a straight line with constant velocity?
$(2)$ In which case will the particle describe a helical path with axis along the positive $z$ direction?
$(3)$ In which case would the particle move in a straight line along the negative direction of $y$-axis (i.e., move along $-\hat{y}$)?

$A$ light charged particle is revolving in a circle of radius $r$ due to the electrostatic attraction of a static heavy particle with an opposite charge. How does the magnetic field $B$ at the centre of the circle, produced by the moving charge, depend on $r$?

There exists a uniform magnetic field of magnitude $1\, T$ and a uniform electric field of magnitude $1\, V/m$ along the positive $y-$ axis. $A$ charged particle of mass $1\, kg$ and charge $1\, C$ has an initial velocity of $1\, m/s$ along the $x-$ axis and is at the origin at $t = 0$. Find the coordinates of the particle at time $t = \pi$ seconds.