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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.
$(B,D)$
$(B,C)$
$(A,D)$
$(C,D)$
Solution
If $\theta=0^{\circ}$ then due to magnetic force path is circular but due to force $qE (\uparrow) q$ will have accelerated motion along $y$-axis. So combined path of $q$ will be a helical path with variable pitch so $(A)$ and $(B)$ are wrong. If $\theta=10^{\circ}$ then due to $v \cos \theta$, path is circular and due to $qE E _0$ and $v \sin \theta, q$ has accelerated motion along $y$-axis so combined path is a helical path with variable pitch $(C)$ is correct.
If $\theta=90^{\circ}$ then $F_B=0$ and due to $q E_0$ motion is accelerated along $y$-axis. $(D)$
