A particle is projected in a plane perpendicular to a uniform magnetic field. The area bounded by the path described by the particle is proportional to

Options

A. the velocity
B. the momentum
C. the kinetic energy
D. none of these

Answer

To solve the problem, we need to analyze the motion of a charged particle projected in a plane perpendicular to a uniform magnetic field. We'll derive the relationship between the area bounded by the path of the particle and its velocity, mass, and charge.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a uniform magnetic field (B) directed into the plane of the paper.
- A charged particle (with charge Q) is projected with an initial velocity (V) perpendicular to this magnetic field.

2. **Determine the Force on the Particle**:
- The magnetic force (F) acting on the charged particle can be expressed using the formula:
\[ F = Q(V \times B) \]
- Since V is perpendicular to B, the magnitude of the magnetic force is:
\[ F = QVB \]

3. **Centripetal Force**:
- The particle will move in a circular path due to the magnetic force acting as the centripetal force. The centripetal force required for circular motion is given by:
\[ F_c = \frac{mv^2}{R} \]
- Where \( m \) is the mass of the particle and \( R \) is the radius of the circular path.

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

5. **Solving for the Radius (R)**:
- Rearranging the equation to solve for R gives:
\[ R = \frac{mv}{QB} \]

6. **Calculating the Area (A)**:
- The area (A) of the circular path traced by the particle is given by:
\[ A = \pi R^2 \]
- Substituting the expression for R:
\[ A = \pi \left(\frac{mv}{QB}\right)^2 \]
- Expanding this gives:
\[ A = \pi \frac{m^2v^2}{Q^2B^2} \]

7. **Proportionality**:
- From the expression for area, we can see that:
\[ A \propto m^2v^2 \]
- This indicates that the area is proportional to the square of the velocity of the particle.

8. **Conclusion**:
- The area bounded by the path described by the particle is proportional to the square of its velocity, which is also related to the kinetic energy of the particle (since kinetic energy \( KE \propto \frac{1}{2} mv^2 \)).
- Therefore, the area is proportional to the kinetic energy of the particle.

### Final Answer:
The area bounded by the path described by the particle is proportional to the kinetic energy of the particle.

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