A charged particle(q) is projected perpendicular to a uniform magnetic field (B) with speed u. A is the area bounded by the path followed by electrons. K is kinetic energy of the particle. Then how will the graph between A and K be?

Options

A. Straight line
B. Parabola
C. Ellipse
D. None of these

Explanation

To solve the problem, we need to analyze the relationship between the area \( A \) bounded by the path of a charged particle moving in a magnetic field and its kinetic energy \( K \).

### Step-by-Step Solution:

1. **Understanding the Motion of the Charged Particle**:
- A charged particle with charge \( q \) and mass \( m \) is projected perpendicular to a uniform magnetic field \( B \) with speed \( u \).
- The magnetic force acting on the particle is given by \( F = q(v \times B) \). Since the velocity \( v \) is perpendicular to the magnetic field \( B \), the magnitude of the magnetic force is \( F = qBu \).

2. **Centripetal Force and Radius of Circular Path**:
- The magnetic force acts as the centripetal force that keeps the particle in circular motion. Thus, we can equate the magnetic force to the centripetal force:
\[ F = \frac{mv^2}{r} \]
- Setting the forces equal gives:
\[ qBu = \frac{mv^2}{r} \]
- Rearranging this equation to find the radius \( r \):
\[ r = \frac{mv}{qB} \]
- Substituting \( v = u \) (the initial speed of the particle):
\[ r = \frac{mu}{qB} \]

3. **Calculating the Area \( A \)**:
- The area \( A \) bounded by the circular path of the particle is given by the formula for the area of a circle:
\[ A = \pi r^2 \]
- Substituting the expression for \( r \):
\[ A = \pi \left(\frac{mu}{qB}\right)^2 = \frac{\pi m^2 u^2}{q^2 B^2} \]

4. **Relating Area \( A \) to Kinetic Energy \( K \)**:
- The kinetic energy \( K \) of the particle is given by:
\[ K = \frac{1}{2} mu^2 \]
- We can express \( u^2 \) in terms of \( K \):
\[ u^2 = \frac{2K}{m} \]
- Substituting \( u^2 \) back into the equation for \( A \):
\[ A = \frac{\pi m^2}{q^2 B^2} \cdot \frac{2K}{m} = \frac{2\pi m}{q^2 B^2} K \]

5. **Conclusion on the Relationship**:
- The relationship between area \( A \) and kinetic energy \( K \) can be expressed as:
\[ A = \left(\frac{2\pi m}{q^2 B^2}\right) K \]
- This indicates a linear relationship between \( A \) and \( K \) (i.e., \( A \) is directly proportional to \( K \)).
- Therefore, the graph between area \( A \) and kinetic energy \( K \) will be a straight line.

### Final Answer:
The graph between area \( A \) and kinetic energy \( K \) is a straight line.