The path of a charged particle in a uniform magnetic field, when the velocity and the magnetic field are perpendicular to each other is a

Options

A. circle
B. parabola
C. helix
D. straight line

Answer

To determine the path of a charged particle in a uniform magnetic field when the velocity and the magnetic field are perpendicular to each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Situation**:
- We have a charged particle with charge \( q \).
- The particle is moving with a velocity \( \vec{v} \) in a uniform magnetic field \( \vec{B} \).
- It is given that \( \vec{v} \) is perpendicular to \( \vec{B} \).

2. **Applying the Lorentz Force Law**:
- The force \( \vec{F} \) acting on the charged particle due to the magnetic field is given by the equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
- This force is perpendicular to both the velocity \( \vec{v} \) and the magnetic field \( \vec{B} \).

3. **Determining the Direction of the Force**:
- Let's assume the velocity \( \vec{v} \) is in the \( \hat{i} \) direction (along the x-axis) and the magnetic field \( \vec{B} \) is in the \( \hat{j} \) direction (along the y-axis).
- The cross product \( \vec{v} \times \vec{B} \) can be calculated:
\[ \vec{F} = q (\hat{i} \times \hat{j}) = q \hat{k} \]
- This indicates that the force \( \vec{F} \) acts in the \( \hat{k} \) direction (along the z-axis).

4. **Analyzing the Motion**:
- Since the magnetic force is always perpendicular to the velocity of the particle, it does not do any work on the particle. Thus, the speed of the particle remains constant.
- The magnetic force acts as a centripetal force, causing the particle to change direction but not speed.

5. **Conclusion about the Path**:
- The motion of the charged particle under the influence of a constant perpendicular magnetic force results in circular motion.
- Therefore, the path of the charged particle is circular.

### Final Answer:
The path of the charged particle in a uniform magnetic field, when the velocity and the magnetic field are perpendicular to each other, is a **circle**.

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