The change in the kinetic energy of a charged particle moving in a uniform magnetic field will be zero when its velocity is

Options

A. Parallel to the magnetic field
B. Perpendicular to the magnetic field
C. At any angle with the magnetic field
D. All of these

Answer

To determine when the change in kinetic energy of a charged particle moving in a uniform magnetic field will be zero, we can analyze the relationship between the velocity of the particle, the magnetic field, and the work done by the magnetic force.

### Step-by-Step Solution:

1. **Understanding Magnetic Force**:
The magnetic force \( \mathbf{F}_B \) acting on a charged particle moving with velocity \( \mathbf{V} \) in a magnetic field \( \mathbf{B} \) is given by the equation:
\[ \mathbf{F}_B = q (\mathbf{V} \times \mathbf{B}) \]
where \( q \) is the charge of the particle. This force is always perpendicular to both the velocity vector \( \mathbf{V} \) and the magnetic field vector \( \mathbf{B} \).

2. **Direction of Force**:
Since the magnetic force is perpendicular to the velocity of the particle, it does not do any work on the particle. Work done \( W \) by a force is given by:
\[ W = \mathbf{F} \cdot \mathbf{d} \]
where \( \mathbf{d} \) is the displacement. If \( \mathbf{F} \) is perpendicular to \( \mathbf{d} \), then:
\[ W = 0 \]

3. **Work-Energy Theorem**:
According to the work-energy theorem, the work done by all forces acting on an object is equal to the change in kinetic energy \( \Delta KE \):
\[ W = \Delta KE \]
Since the work done by the magnetic force is zero, we have:
\[ \Delta KE = 0 \]

4. **Conclusion**:
The change in kinetic energy of the charged particle will be zero regardless of the angle between the velocity vector \( \mathbf{V} \) and the magnetic field vector \( \mathbf{B} \). Therefore, the answer is that the change in kinetic energy will be zero at any angle with the magnetic field.

### Final Answer:
The change in kinetic energy of a charged particle moving in a uniform magnetic field will be zero when its velocity is at any angle with the magnetic field.