A proton of mass m and charge q enters a region of uniform magnetic field of a magnitude B with a velocity v directed perpendicular to the magnetic field. It moves in a circular path and leaves the magnetic field after completing a quarter of a circle. The time spent by the proton inside the magnetic field is proportional to:

Options

A. `v^(-1//2)`
B. `v^(1//2)`
C. `v^(0)`
D. v

Explanation

To solve the problem, we need to analyze the motion of a proton in a magnetic field. The proton enters a uniform magnetic field with a velocity perpendicular to the field and moves in a circular path. We are asked to find the time spent by the proton inside the magnetic field, which is proportional to certain variables.

### Step-by-Step Solution:

1. **Understanding the Motion in a Magnetic Field**:
When a charged particle (like a proton) moves in a magnetic field perpendicular to its velocity, it experiences a magnetic force that acts as a centripetal force, causing it to move in a circular path.

2. **Centripetal Force and Magnetic Force**:
The magnetic force \( F \) acting on the proton is given by:
\[ F = qvB \]
where:
- \( q \) is the charge of the proton,
- \( v \) is the velocity of the proton,
- \( B \) is the magnetic field strength.

This magnetic force provides the necessary centripetal force to keep the proton in circular motion:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the proton and \( r \) is the radius of the circular path.

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

4. **Solving for the Radius**:
Rearranging the equation to find the radius \( r \):
\[ r = \frac{mv}{qB} \]

5. **Distance Traveled in the Magnetic Field**:
The proton completes a quarter of a circle while in the magnetic field. The distance \( d \) it travels is:
\[ d = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

6. **Substituting for Radius**:
Substitute the expression for \( r \):
\[ d = \frac{\pi}{2} \times \frac{mv}{qB} \]

7. **Time Spent in the Magnetic Field**:
The time \( t \) spent in the magnetic field can be calculated using the formula:
\[ t = \frac{d}{v} = \frac{\frac{\pi}{2} \times \frac{mv}{qB}}{v} = \frac{\pi m}{2qB} \]

8. **Proportionality**:
From the final expression for time \( t \):
\[ t \propto \frac{m}{qB} \]
This shows that the time spent by the proton inside the magnetic field is proportional to the mass \( m \) of the proton and inversely proportional to the charge \( q \) and the magnetic field strength \( B \).

### Final Answer:
The time spent by the proton inside the magnetic field is proportional to \( \frac{m}{qB} \).