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A particle mass m, charge Q, and kinetic energy T enter a transverse uniform magnetic field of induction B⃗\vec{B}. After 3 sec the kinetic energy of the particle will be:

A

3T

B

2T

C

T

D

4T

Step-by-Step Solution

  1. Nature of Magnetic Force: The magnetic force exerted on a moving charge is given by the Lorentz force formula: F⃗=q(v⃗×B⃗)\vec{F} = q(\vec{v} \times \vec{B}). By definition of the cross product, this force is always perpendicular to the velocity vector (v⃗\vec{v}) of the particle .
  2. Work Done: Since the force is perpendicular to the instantaneous displacement (dl⃗=v⃗dtd\vec{l} = \vec{v}dt), the work done by the magnetic force is zero: W=∫F⃗⋅dl⃗=∫F⃗⋅v⃗dt=0W = \int \vec{F} \cdot d\vec{l} = \int \vec{F} \cdot \vec{v} dt = 0 .
  3. Work-Energy Theorem: According to the work-energy theorem, the change in kinetic energy is equal to the work done. Since W=0W = 0, the change in kinetic energy ΔK=0\Delta K = 0.
  4. Conclusion: The speed and kinetic energy of a charged particle moving in a magnetic field remain constant, even though its direction changes. Thus, the kinetic energy remains TT after any time interval .
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