Consider the motion of a positive point charge in a region where there are simultaneous uniform eletric and magnetic field `E=E_0hatj` and `B=B_0hatj`. 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 `tgt0?`
a. If `theta=0^0` the charge moves in a circular path in the xy-plane.
`b.If theta=0^0` the charge undergoes helical motion with constant pitch along the y-axis.
c. If `theta=1 0^0` the charge undergoes helicalmotion with its pitch increasing with time along the `y`-axis.
d. If `theta=90^@` the charge undergoes linear but accelerated motion along the `y`-axis.

Options

A. If `theta=0^(@)`, the charge moves in a circular path in the x-z plane
B. If `theta=0^(@)`, the charge undergoes helical motion with constant pitch along the y-axis
C. If `theta=10^(@)`, the charge undergoes helical motion with its pitch increasing with time, along the y-axis
D. If `theta=90^(@)` the charge undergoes linear but accelerated motion along the y-axis

Answer

C, D

Explanation

(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_(_|_)` and B 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.
(D) Magnetic force is zero, as 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).