If a proton enters perpendicularly a magnetic field with velocity v, then time period of revolution is T. If proton enters with velocity 2v, then time period will be

Options

A. T
B. 2T
C. 3T
D. 4T

Answer

To solve the problem, we need to analyze the relationship between the time period of revolution of a proton in a magnetic field and its velocity.

### Step-by-Step Solution:

1. **Understanding the Motion of a Charged Particle in a Magnetic Field**:
When a charged particle, like a proton, enters a magnetic field perpendicularly, it experiences a magnetic force that causes it to move in a circular path. The magnetic force (Lorentz force) is given by:
\[ F = q(\mathbf{v} \times \mathbf{B}) \]
where \( q \) is the charge of the proton, \( \mathbf{v} \) is its velocity, and \( \mathbf{B} \) is the magnetic field.

2. **Centripetal Force**:
The magnetic force acts as the centripetal force required to keep the proton in circular motion. The centripetal force is given by:
\[ F = \frac{mv^2}{R} \]
where \( m \) is the mass of the proton, \( v \) is its velocity, and \( R \) is the radius of the circular path.

3. **Equating Forces**:
Setting the magnetic force equal to the centripetal force, we have:
\[ qvB = \frac{mv^2}{R} \]

4. **Solving for Radius**:
Rearranging the equation gives:
\[ R = \frac{mv}{qB} \]

5. **Time Period of Revolution**:
The time period \( T \) for one complete revolution is given by:
\[ T = \frac{2\pi R}{v} \]
Substituting the expression for \( R \):
\[ T = \frac{2\pi}{v} \cdot \frac{mv}{qB} = \frac{2\pi m}{qB} \]
This shows that the time period \( T \) is independent of the velocity \( v \).

6. **Considering the New Velocity**:
If the proton enters the magnetic field with a new velocity \( 2v \), we can analyze the time period again:
\[ T' = \frac{2\pi m}{qB} \]
Since the expression for \( T \) does not include \( v \), the time period remains the same.

### Conclusion:
The time period \( T' \) when the proton enters with velocity \( 2v \) is still:
\[ T' = T \]
Thus, the time period remains unchanged.

### Final Answer:
The time period will be \( T \).

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