A beam of protons enters a uniform magnetic field of 0.3T with velocity of `4xx10^5m//s` in a direction making an angle of `60^(@)` with the direction of magnetic field. The path of motion of the particle will be

Options

A. circular
B. Straight line
C. Spiral
D. helical

Answer

To solve the problem of the motion of a beam of protons entering a uniform magnetic field, we need to analyze the components of velocity and their effects on the motion of the protons.

### Step-by-step Solution:

1. **Identify Given Values**:
- Magnetic field strength, \( B = 0.3 \, \text{T} \)
- Velocity of protons, \( v = 4 \times 10^5 \, \text{m/s} \)
- Angle with the magnetic field, \( \theta = 60^\circ \)

2. **Resolve the Velocity into Components**:
- The velocity can be resolved into two components:
- **Parallel Component**: \( v_{\parallel} = v \cos(\theta) \)
- **Perpendicular Component**: \( v_{\perpendicular} = v \sin(\theta) \)

- Calculate these components:
- \( v_{\parallel} = 4 \times 10^5 \cos(60^\circ) = 4 \times 10^5 \times \frac{1}{2} = 2 \times 10^5 \, \text{m/s} \)
- \( v_{\perpendicular} = 4 \times 10^5 \sin(60^\circ) = 4 \times 10^5 \times \frac{\sqrt{3}}{2} = 2 \sqrt{3} \times 10^5 \, \text{m/s} \)

3. **Analyze the Motion**:
- The component of velocity parallel to the magnetic field (\( v_{\parallel} \)) does not experience any magnetic force and will continue to move in a straight line.
- The component of velocity perpendicular to the magnetic field (\( v_{\perpendicular} \)) will experience a magnetic force, causing it to move in a circular path.

4. **Determine the Path of Motion**:
- Since the protons have a component of motion that is circular (due to the perpendicular component) and a component that is linear (due to the parallel component), the overall path of the protons will be a helical path.

5. **Conclusion**:
- Therefore, the path of motion of the protons in the magnetic field will be **helical**.

### Final Answer:
The path of motion of the particle will be **helical**.

---