Velocity of a charged particle can remain unchanged. If

Options

A. it is moving only in electric field
B. it is moving only in magnetic field
C. it is moving both in electric and magnetic fields
D. neither in electric nor in magnetic fields

Answer

To determine under what conditions the velocity of a charged particle can remain unchanged, let's analyze each of the given statements step by step.

### Step-by-Step Solution:

1. **Statement A: Moving in an Electric Field**
- When a charged particle moves in an electric field, it experiences a force given by \( F = qE \), where \( q \) is the charge and \( E \) is the electric field strength.
- This force causes the particle to accelerate according to Newton's second law, \( F = ma \).
- Since there is a non-zero force acting on the particle, it will experience acceleration, which means its velocity will change.
- **Conclusion**: This statement is **false**.

2. **Statement B: Moving in a Magnetic Field**
- When a charged particle moves in a magnetic field, it experiences a magnetic force given by \( F = q(v \times B) \), where \( v \) is the velocity of the particle and \( B \) is the magnetic field.
- The magnetic force acts perpendicular to both the velocity of the particle and the magnetic field direction.
- Since the force is always perpendicular to the displacement, the work done by the magnetic force is zero (as work is calculated as \( W = F \cdot d \cos(\theta) \), and here \( \theta = 90^\circ \)).
- Because no work is done, the kinetic energy remains constant, and hence the speed (magnitude of velocity) of the particle remains unchanged.
- **Conclusion**: This statement is **true**.

3. **Statement C: Moving in Both Electric and Magnetic Fields**
- If a charged particle is moving in both electric and magnetic fields, the forces due to these fields can act in such a way that they cancel each other out.
- If the electric force \( F_E = qE \) and the magnetic force \( F_B = q(v \times B) \) are equal in magnitude and opposite in direction, the net force on the particle will be zero.
- With no net force acting on the particle, it will not accelerate, and its velocity will remain unchanged.
- **Conclusion**: This statement is **true**.

4. **Statement D: Neither in Electric nor in Magnetic Field**
- If a charged particle is not in an electric or magnetic field, there are no forces acting on it.
- According to Newton's first law of motion, an object in motion will remain in motion with the same velocity unless acted upon by a net external force.
- Therefore, the velocity of the charged particle will remain unchanged.
- **Conclusion**: This statement is **true**.

### Final Conclusion:
- The statements that are true are **B, C, and D**.
- The only statement that is false is **A**.