Question

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`?
A. If ` theta = 0@` , the charge moves in a circular path in the ` x-z` plane.
B. If ` theta = 0@`, the charge undergoes helical path 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 1

Correct Answer - C::D
When ` theta = 0^(@), the charged particle is projected along ` x - axis` , due to `vec(B)` the charged particle will tend to move in a circular path in ` y -z `plane but due to force of electric field , the particle will move in a helical path with increasing pitch. Therfore options `(A) and (B)` are incorrect.
When ` theta = 10^(@)`, we can resolve velocity into two rectangular components. One al,ong ` x- axis(v cos 10^(@))` and one along ` y - axis (v sin 10^(@))`. Due to ` v cos 10@` , the particle will move in circular path and due to ` v sin 10^(@)` plus the force due to electric field, the particle will undergo helical motion with its pitch increasing.
If ` theta = 90^(@)`, the charge is moving along the magnetic field. Therefore the force due to magnetic field is zero . But the force due to electric field will accelerate the particle along ` y - axis`.