Charged Particle in Magnetic Field (2003)

Question: A charged particle moves through a magnetic field in a direction perpendicular to it. Then the

(a) speed of the particle remains unchanged

(b) direction of the particle remains unchanged

(c) acceleration remains unchanged

(d) velocity remains unchanged

Detailed Explanation

When a charged particle enters a magnetic field perpendicularly, it experiences a magnetic force given by: \[ \vec{F} = q (\vec{v} \times \vec{B}), \] where \( q \) is the charge, \( \vec{v} \) is the velocity, and \( \vec{B} \) is the magnetic field. Since the velocity is perpendicular to the magnetic field, the force is perpendicular to both \( \vec{v} \) and \( \vec{B} \).

This perpendicular force causes the particle to move in a circular path. Let’s analyze each option:

• (a) Speed remains unchanged:
  The magnetic force is always perpendicular to the velocity. Since work done is \( W = \vec{F} \cdot \vec{d} \), and the force and displacement (along the velocity) are perpendicular, no work is done. Thus, the kinetic energy, \( K = \frac{1}{2}mv^2 \), remains constant. With mass \( m \) constant, the speed \( v \) remains unchanged. This option is correct.
• (b) Direction remains unchanged:
  The force causes the particle to follow a circular path, meaning its direction changes continuously. This option is incorrect.
• (c) Acceleration remains unchanged:
  The acceleration is centripetal, given by \( a = \frac{v^2}{r} \), where \( r \) is the radius of the path. Its magnitude is constant because speed \( v \) is constant, but its direction changes as the particle moves, always pointing toward the center of the circle. Since acceleration is a vector, and its direction changes, it does not remain unchanged. This option is incorrect.
• (d) Velocity remains unchanged:
  Velocity is a vector with magnitude (speed) and direction. While the speed remains constant, the direction changes due to circular motion. Thus, the velocity vector changes, making this option incorrect.

Conclusion: The only quantity that remains unchanged is the speed of the particle.

Final Answer: Option (a) speed of the particle remains unchanged.