A soldier directs a laser beam on an enemy by reflecting the beam from a mirror. If the mirror is rotated by an angle `theta`, by what angle will the reflected beam rotate?

Options

A. `theta/2`
B. `theta`
C. `2theta`
D. None of these

Answer

To solve the problem of how much the reflected beam rotates when the mirror is rotated by an angle \( \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a mirror that reflects a laser beam directed towards an enemy. The incident ray strikes the mirror at an angle \( \alpha \) with respect to the normal (perpendicular) to the mirror surface.

2. **Applying the Law of Reflection**:
- According to the law of reflection, the angle of incidence (the angle between the incident ray and the normal) is equal to the angle of reflection (the angle between the reflected ray and the normal). Therefore, both the incident and reflected angles are \( \alpha \).

3. **Rotating the Mirror**:
- When the mirror is rotated by an angle \( \theta \), the normal to the mirror also rotates by the same angle \( \theta \). This changes the orientation of the normal line.

4. **New Angles After Rotation**:
- After the mirror is rotated, the new angle of incidence remains \( \alpha \) (since the incident ray does not change), but the new angle of reflection will now be measured from the new normal. Thus, the angle of reflection becomes \( \alpha + \theta \).

5. **Calculating the Change in the Reflected Beam**:
- The change in the angle of the reflected beam can be calculated. Initially, the reflected ray made an angle \( \alpha \) with the reference line (the original position of the reflected ray). After the rotation, the reflected ray makes an angle \( \alpha + 2\theta \) with the reference line.

6. **Finding the Angle of Rotation**:
- The angle of rotation of the reflected beam \( \beta \) can be calculated as:
\[ \beta = (\alpha + 2\theta) - \alpha = 2\theta \]

### Final Answer:
The angle by which the reflected beam rotates is \( 2\theta \).

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