A particle of mass m and charge q is accelerated by a potential difference V volt and made to enter a magnetic field region at an angle `theta` with the field. At the same moment, another particle of same mass and charge is projected in the direction of the field from the same point. Magnetic field induction is B. What would be the speed of second particle so that both particles meet again and again after regular interval of time. Also, find the time interval after which they meet and the distance travelled by the second particle during that interval.

Options

A. `(2pim)/(qB)`
B. `(pim)/(2qB)`
C. `(pim)/(qB)`
D. `(3pim)/(2qB)`

Explanation

To solve the problem, we need to determine the speed of the second particle, the time interval after which both particles meet, and the distance traveled by the second particle during that interval. Let's break it down step by step.

### Step 1: Calculate the speed of the first particle (v1)
The first particle is accelerated by a potential difference \( V \). The kinetic energy gained by the particle is equal to the work done on it by the electric field:

\[ qV = \frac{1}{2} mv_1^2 \]

Rearranging this equation to solve for \( v_1 \):

\[ v_1 = \sqrt{\frac{2qV}{m}} \]

### Step 2: Determine the speed of the second particle (v2)
The second particle is projected in the direction of the magnetic field. To ensure that both particles meet again and again, the horizontal component of the velocity of the first particle (which moves at an angle \( \theta \)) must equal the velocity of the second particle.

The horizontal component of the velocity of the first particle is:

\[ v_{1x} = v_1 \cos \theta \]

Thus, we set \( v_2 \) equal to \( v_{1x} \):

\[ v_2 = v_1 \cos \theta \]

Substituting the expression for \( v_1 \):

\[ v_2 = \sqrt{\frac{2qV}{m}} \cos \theta \]

### Step 3: Find the time interval (T) after which they meet
The time period \( T \) for the motion of the first particle in the magnetic field can be calculated using the formula:

\[ T = \frac{2\pi m}{qB} \]

### Step 4: Calculate the distance traveled by the second particle during the time interval T
The distance traveled by the second particle during the time interval \( T \) is given by:

\[ \text{Distance} = v_2 \cdot T \]

Substituting the expression for \( v_2 \):

\[ \text{Distance} = \left(\sqrt{\frac{2qV}{m}} \cos \theta\right) \cdot \left(\frac{2\pi m}{qB}\right) \]

Simplifying this expression gives:

\[ \text{Distance} = \frac{2\pi m}{qB} \cdot \sqrt{\frac{2qV}{m}} \cos \theta \]

### Final Results
1. **Speed of the second particle**:
\[ v_2 = \sqrt{\frac{2qV}{m}} \cos \theta \]

2. **Time interval after which they meet**:
\[ T = \frac{2\pi m}{qB} \]

3. **Distance traveled by the second particle during that interval**:
\[ \text{Distance} = \frac{2\pi m}{qB} \cdot \sqrt{\frac{2qV}{m}} \cos \theta \]