Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E}=E_0 \hat{j}$ and $\vec{B}=B_0 \hat{j}$. At time $t=0$, this charge has velocity $\nabla$ in the $x$-y plane, making an angle $\theta$ with $x$-axis. Which of the following option$(s)$ is(are) correct for time $t>0$ ?

$(A)$ If $\theta=0^{\circ}$, the charge moves in a circular path in the $x-z$ plane.

$(B)$ If $\theta=0^{\circ}$, the charge undergoes helical motion with constant pitch along the $y$-axis.

$(C)$ If $\theta=10^{\circ}$, the charge undergoes helical motion with its pitch increasing with time, along the $y$-axis.

$(D)$ If $\theta=90^{\circ}$, the charge undergoes linear but accelerated motion along the $y$-axis.

In toroid magnetic field on axis will be the radius $=0.5\, cm ,$ current $=1.5\, A ,$ turns $=250,$ permeability $=700$ (in Tesla)

A charged particle enters a uniform magnetic field with velocity vector making an angle of $30^o$ with the magnetic field. The particle describes a helical trajectory of pitch $x$ . The radius of the helix is

A charged particle enters a magnetic field $H$ with its initial velocity making an angle of $45^\circ $ with $H$. The path of the particle will be

A proton of mass $1.67\times10^{-27}\, kg$ and charge $1.6\times10^{-19}\, C$ is projected with a speed of $2\times10^6\, m/s$ at an angle of $60^o$ to the $X-$ axis. If a uniform magnetic field of $0.104\, tesla$ is applied along the $Y-$ axis, the path of the proton is

A proton is projected with a velocity $10^7\, m/s$, at right angles to a uniform magnetic field of induction $100\, mT$. The time (in second) taken by the proton to traverse $90^o$ arc is  $(m_p = 1.65\times10^{-27}\, kg$ and $q_p = 1.6\times10^{-19}\, C)$