Question

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields $\vec{E}=E_0 \widehat{j}$ and $\vec{B}=B_0\widehat{j}$. At time t = 0, this charge has velocity v in the x-y plane, making an angle $\theta$ with the 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 $\theta=0^\circ$, the charge undergoes helical motion with its pitch increasing with time, along the y-axis.
• D. If $\theta=0^\circ$, the charge undergoes linear but accelerated motion along the y-axis.

Answer 1

Answer: (C), (D)

(a) and (b)
Magnetic field will rotate the particle in a circular path (in x-z plane or perpendicular to B). Electric field will exert a constant force on the particle in positive y-direction. Therefore, resultant path is neither purely circular nor helical or the options (a) and (b) both are wrong.
$(c) v_{\perp}$ will rotate the particle in a circular path in x- z plane (or perpendicular to B). Further$v_{||}$ and E will move the particle (with increasing speed) along positive y-axis (or along the axis of above circular path). Therefore, the resultant path is helical with increasing pitch, along the y-axis (or along B and E). Therefore option (c) is correct.
Magnetic force is zero, as $\theta$ between B and v is zero. But electric force will act in y-direction. Therefore, motion is
1-D and uniformly accelerated (towards positive y-direction).