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. (a)`v^(-1//2)`
B. (b)`v^(1//2)`
C. (c)`v^(0)`
D. (d)v

Explanation

To solve the problem, we need to analyze the motion of a proton moving in a magnetic field. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Motion of the Proton
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 radius of this circular path can be determined using the formula:

\[ r = \frac{mv}{qB} \]

where:
- \( r \) is the radius of the circular path,
- \( m \) is the mass of the proton,
- \( v \) is the velocity of the proton,
- \( q \) is the charge of the proton,
- \( B \) is the magnetic field strength.

### Step 2: Determine the Distance Traveled
Since the proton completes a quarter of a circle, the distance \( d \) it travels while in the magnetic field is:

\[ d = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

### Step 3: Substitute for Radius
Substituting the expression for \( r \) into the distance formula gives:

\[ d = \frac{\pi}{2} \left(\frac{mv}{qB}\right) \]

### Step 4: Calculate the Time Spent in the Magnetic Field
The time \( t \) spent by the proton in the magnetic field can be calculated using the formula:

\[ t = \frac{d}{v} \]

Substituting the expression for \( d \):

\[ t = \frac{\frac{\pi}{2} \left(\frac{mv}{qB}\right)}{v} \]

### Step 5: Simplify the Expression
Now, simplifying the expression for \( t \):

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

### Step 6: Analyze Proportionality
From the final expression, we can see that the time \( t \) is proportional to \( m \) (mass of the proton), \( 1/q \) (inverse of charge), and \( 1/B \) (inverse of the magnetic field strength). Importantly, it does not depend on the velocity \( v \).

### Conclusion
Thus, we conclude that the time spent by the proton inside the magnetic field is proportional to:

\[ t \propto \frac{m}{qB} \]

Since it does not depend on \( v \), we can say that it is proportional to \( v^0 \) (which equals 1).

### Final Answer
The correct option is **C**.

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