A particle of mass m and charge q is moving in a region where uniform, constant electric and mangetic fields `vec E and vec B` are present. `vec E and vec B` are parallel to each other. At time `t=0,` the velocity `vec v_0` of the particle is perpendicular to `vec E` (Assume that its speed is always `lt lt c`, the speed of light in vacuum). Find the velocity `vec v` of the particle at time `t`. You must express your answer in terms of `t, q, m,` the vector `vec v_0, vec E` and `vec B` and their magnitudes `vec v_0, vec E` and `vec B`.

Explanation

To solve the problem, we need to analyze the motion of a charged particle in the presence of both electric and magnetic fields. Given that the electric field \(\vec{E}\) and magnetic field \(\vec{B}\) are parallel, and the initial velocity \(\vec{v_0}\) of the particle is perpendicular to \(\vec{E}\), we can follow these steps:

### Step 1: Identify Forces Acting on the Particle
The particle experiences two forces:
1. **Electric Force**: \(\vec{F_E} = q \vec{E}\)
2. **Magnetic Force**: \(\vec{F_B} = q (\vec{v} \times \vec{B})\)

### Step 2: Determine the Acceleration Due to Electric Field
The acceleration of the particle due to the electric field is given by:
\[ \vec{a_E} = \frac{\vec{F_E}}{m} = \frac{q \vec{E}}{m} \]

### Step 3: Calculate the Velocity in the Electric Field Direction
Since the initial velocity \(\vec{v_0}\) has no component in the direction of \(\vec{E}\) (which is along the y-axis), the velocity in the y-direction at time \(t\) can be determined using:
\[ v_y(t) = v_{y0} + a_E t = 0 + \left(\frac{qE}{m}\right)t = \frac{qE}{m} t \]

### Step 4: Analyze the Motion in the Magnetic Field
The magnetic force acts perpendicular to the velocity of the particle. The particle will undergo circular motion in the plane perpendicular to \(\vec{B}\). The angular frequency \(\omega\) of this circular motion is given by:
\[ \omega = \frac{qB}{m} \]

### Step 5: Express the Velocity Components
The velocity components can be expressed as:
- **In the x-direction** (due to the circular motion):
\[ v_x(t) = v_{0x} \cos(\omega t) = v_{0} \cos\left(\frac{qB}{m} t\right) \]
- **In the z-direction** (due to the circular motion):
\[ v_z(t) = v_{0} \sin\left(\frac{qB}{m} t\right) \]

### Step 6: Combine the Velocity Components
The total velocity vector \(\vec{v}\) of the particle at time \(t\) can thus be expressed as:
\[ \vec{v}(t) = v_x(t) \hat{i} + v_y(t) \hat{j} + v_z(t) \hat{k} \]
Substituting the expressions we found:
\[ \vec{v}(t) = v_{0} \cos\left(\frac{qB}{m} t\right) \hat{i} + \left(\frac{qE}{m} t\right) \hat{j} + v_{0} \sin\left(\frac{qB}{m} t\right) \hat{k} \]

### Final Expression
Thus, the velocity of the particle at time \(t\) is:
\[ \vec{v}(t) = v_{0} \cos\left(\frac{qB}{m} t\right) \hat{i} + \left(\frac{qE}{m} t\right) \hat{j} + v_{0} \sin\left(\frac{qB}{m} t\right) \hat{k} \]