Question

A charged particle moving with velocity $\vec{v}$ enters a region of uniform magnetic field $\vec{B}$. The force acting on it is non-zero. Would the particle gain any energy? Explain.

Hint:

The magnetic force is perpendicular to the velocity, so it does zero work. The kinetic energy and speed stay constant.

Answer 1

Step 1: Understanding the Concept:
A charged particle with charge \(q\) moving with velocity \(\vec{v}\) in a magnetic field \(\vec{B}\) feels the magnetic force:
\[ \vec{F} = q(\vec{v} \times \vec{B}) \]

Step 2: Direction of the force:
The cross product is perpendicular to both \(\vec{v}\) and \(\vec{B}\). So the force is always at right angles to the velocity.

Step 3: Work done:
The rate of work done by the force is:
\[ P = \vec{F} \cdot \vec{v} = q(\vec{v} \times \vec{B}) \cdot \vec{v} = 0 \]
The work done is zero, so the kinetic energy does not change by the work-energy theorem. The force only changes the direction of the velocity, not its magnitude.

Final Answer:
No, the particle does not gain energy. Its speed and kinetic energy stay constant, even though the force is non-zero.