A proton is moving perpendicular to the magnetic field with speed v and it is found that the proton takes T time to complete its circular path once. If the speed of the particle is increased to 2v, then how much time will it take to complete the circle?

Options

A. T
B. `T//2 `
C. `2 T`
D. `4 T`

Explanation

To solve the problem, we will analyze the motion of a proton in a magnetic field and how its time period is affected by changes in speed.

### Step-by-Step Solution:

1. **Understanding the Motion of the Proton**:
- A proton is moving perpendicular to a magnetic field with speed \( v \).
- The magnetic force acting on the proton is given by \( F = qvB \), where \( q \) is the charge of the proton, \( v \) is its speed, and \( B \) is the magnetic field strength.

2. **Centripetal Force Requirement**:
- The magnetic force provides the centripetal force required for circular motion. Thus, we have:
\[ F = \frac{mv^2}{r} \]
- Setting the magnetic force equal to the centripetal force gives:
\[ qvB = \frac{mv^2}{r} \]

3. **Finding the Radius of the Circular Path**:
- Rearranging the equation, we can express the radius \( r \) as:
\[ r = \frac{mv}{qB} \]

4. **Calculating the Time Period**:
- The time period \( T \) for one complete revolution is given by the distance traveled in one revolution divided by the speed:
\[ T = \frac{2\pi r}{v} \]
- Substituting for \( r \):
\[ T = \frac{2\pi \left(\frac{mv}{qB}\right)}{v} = \frac{2\pi m}{qB} \]
- Notice that \( T \) does not depend on the speed \( v \).

5. **Increasing the Speed**:
- If the speed of the proton is increased to \( 2v \), we can calculate the new time period \( T' \):
\[ T' = \frac{2\pi r'}{2v} \]
- However, since the radius \( r' \) will also change when speed changes, we need to find the new radius:
\[ r' = \frac{m(2v)}{qB} = 2\left(\frac{mv}{qB}\right) = 2r \]
- Substituting \( r' \) back into the time period equation:
\[ T' = \frac{2\pi (2r)}{2v} = \frac{2\pi r}{v} = T \]

6. **Conclusion**:
- The time period \( T' \) remains the same as \( T \) when the speed is doubled. Therefore, the time taken to complete the circular path does not change with the increase in speed.

### Final Answer:
The time taken to complete the circle when the speed is increased to \( 2v \) is the same as the original time \( T \).