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. `sqrt((qV)/m) cos theta`
B. `sqrt((2qV)/m) cos theta`
C. `sqrt((qV)/m) sin theta`
D. `sqrt((qV)/(2m)) cos theta`

Explanation

To solve the problem step-by-step, we will follow the outlined procedure based on the information given in the question and the video transcript.

### Step 1: Determine the speed of the first particle (U1)
A particle of mass \( m \) and charge \( q \) 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, which is given by:

\[ \frac{1}{2} m U_1^2 = qV \]

From this equation, we can solve for \( U_1 \):

\[ U_1^2 = \frac{2qV}{m} \]

\[ U_1 = \sqrt{\frac{2qV}{m}} \]

### Step 2: Determine the speed of the second particle (U2)
The second particle is projected in the direction of the magnetic field. For both particles to meet again and again, the component of the velocity of the first particle in the direction of the magnetic field must equal the speed of the second particle. The component of \( U_1 \) in the direction of the magnetic field is:

\[ U_1 \cos(\theta) \]

Thus, we set \( U_2 \) equal to this component:

\[ U_2 = U_1 \cos(\theta) \]

Substituting the expression for \( U_1 \):

\[ U_2 = \sqrt{\frac{2qV}{m}} \cos(\theta) \]

### Step 3: Find the time interval (T) after which they meet
The time interval after which the first particle returns to the same position in its helical path can be calculated using the formula for the period of motion in a magnetic field:

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

This is the time taken for the first particle to complete one full cycle in the magnetic field.

### Step 4: Calculate the distance traveled by the second particle during this interval
The distance traveled by the second particle during this time interval \( T \) can be calculated using the formula:

\[ \text{Distance} = U_2 \times T \]

Substituting the values of \( U_2 \) and \( T \):

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

Simplifying this expression:

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

\[ \text{Distance} = \frac{2\pi \sqrt{2mV} \cos(\theta)}{B \sqrt{q}} \]

### Final Results
1. The speed of the second particle \( U_2 \):
\[ U_2 = \sqrt{\frac{2qV}{m}} \cos(\theta) \]

2. The time interval \( T \):
\[ T = \frac{2\pi m}{qB} \]

3. The distance traveled by the second particle during this interval:
\[ \text{Distance} = \frac{2\pi \sqrt{2mV} \cos(\theta)}{B \sqrt{q}} \]