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. a.If `theta=0^0` the charge moves in a circular path in the xy-plane.

B. b. If theta=0^0` the charge undergoes helical motion with constant pitch along the y-axis.

C. c.
If `theta=10^0` the charge undergoes helicalmotion with its pitch increasing with time along the `y`-axis.

D. d. If `theta=90^@` the charge undergoes linear but accelerated motion along the `y`-axis.

Explanation

To solve the problem, we need to analyze the motion of a positive point charge in the presence of uniform electric and magnetic fields. The electric field \( \mathbf{E} = E_0 \hat{j} \) and the magnetic field \( \mathbf{B} = B_0 \hat{j} \) are both directed along the y-axis. The charge has an initial velocity \( \mathbf{v} \) in the x-y-plane, making an angle \( \theta \) with the x-axis.

Let's analyze each case step by step:

### Step 1: Understanding the Forces Acting on the Charge
The force acting on the charge due to the electric field is given by:
\[
\mathbf{F}_E = q \mathbf{E} = q E_0 \hat{j}
...