Consider the motion of a positive point charge in a region where area 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 ` vec(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 gt 0`?

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

Here, `vecE=E_0hatj` and `vecB=B_0hatj`.
Let m, q be the mass and charge of the positive point charge which is moving with velocity `vecv` in the electric and magnetic fields acting along Y-axis.

Resolving `vecv` into two rectangular components we have, `v cos theta` along X-axis and `vsin theta` acts along Y-axis.
Due to component velocity `vsintheta`, the charged particle is accelerated due to electric field with acceleration, `a_y=E_0q//m` along Y-axis.
The particle does not experience any force due to magnetic field.
Due to component velocity `vcostheta`, the particle is accelerated due to electric field with acceleration `a_y=E_0 q//m` along Y-axis.
The particle experiences maximum force due to magnetic field and describes a circular path with time period, `T=2pim//Bq` which is independent of r and v.
Due to combined electric and magnetic fields along Y-axis, when `theta=10^@`, the particle will describe a helical path whose pitch increases with time along the Y-axis (i.e., option (c) is true). When `theta=90^@`, the particle does not experience any force due to magnetic field. Hence it is accelerated along y-axis due to electric field alone. Thus option (d) is true.