Show that a ray of light reflected from a plane mirror rotates througha an angle `2theta` when the
mirror is rotated through an angle `theta` about its axis perpendicular to both the incident ray and
the normal to the surface.

Answer

To show that a ray of light reflected from a plane mirror rotates through an angle of \(2\theta\) when the mirror is rotated through an angle of \(\theta\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Setup**:
- Consider a plane mirror with a normal line (perpendicular to the surface of the mirror) at point of incidence.
- Let the incident ray make an angle \(I\) with the normal. According to the law of reflection, the reflected ray will also make an angle \(R\) with the normal, where \(I = R\).

2. **Rotate the Mirror**:
- Now, we rotate the mirror by an angle \(\theta\) about its axis, which is perpendicular to both the incident ray and the normal to the surface.
- This rotation will also cause the normal to rotate by the same angle \(\theta\).

3. **Determine the New Angles**:
- After the mirror is rotated, the new normal (let's call it normal 2) will be at an angle \(\theta\) from the original normal.
- The angle of incidence with respect to the new normal will now be \(I' = I + \theta\).

4. **Apply the Law of Reflection**:
- Since the angle of incidence equals the angle of reflection, the new angle of reflection \(R'\) will also be \(R' = I' = I + \theta\).

5. **Calculate the Total Rotation of the Reflected Ray**:
- The original angle of reflection was \(R\) and the new angle of reflection is \(R' = I + \theta\).
- The change in the angle of the reflected ray is given by:
\[ \text{Change in angle} = R' - R = (I + \theta) - R \]
- Since \(I = R\), we can substitute \(R\) into the equation:
\[ \text{Change in angle} = (I + \theta) - I = \theta \]
- However, we need to account for the total rotation. The total angle rotated by the reflected ray is:
\[ \text{Total rotation} = R' - R + R - R = \theta + \theta = 2\theta \]

6. **Conclusion**:
- Thus, the reflected ray rotates through an angle of \(2\theta\) when the mirror is rotated through an angle of \(\theta\).

### Final Statement:
Therefore, we have shown that a ray of light reflected from a plane mirror rotates through an angle \(2\theta\) when the mirror is rotated through an angle \(\theta\).

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