A narrow beam of light after reflection by a plane mirror falls on a scale at a distance 100 cm from the mirror. When the mirror is rotated a little, the light spot moves through 2 cm. The angle through which the mirror is rotated is

Options

A. 0.02 rad
B. 0.01 rad
C. 200 rad
D. 0.01/188 `pi ` rad

Answer

To solve the problem, we need to find the angle through which the mirror is rotated when a narrow beam of light reflects off a plane mirror and the light spot moves on a scale. Here’s a step-by-step solution:

### Step 1: Understand the setup
- A plane mirror reflects a narrow beam of light.
- The distance from the mirror to the scale is given as 100 cm.
- When the mirror is rotated slightly, the light spot moves 2 cm on the scale.

### Step 2: Identify the relationship between the arc and the angle
- The movement of the light spot on the scale can be treated as an arc of a circle, where the radius of the circle is the distance from the mirror to the scale (100 cm).
- The formula relating the angle (in radians) to the arc length and radius is:
\[ \theta = \frac{\text{arc}}{\text{radius}} \]

### Step 3: Substitute the known values
- Here, the arc length (the movement of the light spot) is 2 cm, and the radius is 100 cm.
- Substitute these values into the formula:
\[ \theta = \frac{2 \text{ cm}}{100 \text{ cm}} = 0.02 \text{ radians} \]

### Step 4: Relate the angle of mirror rotation to the angle of reflection
- For a plane mirror, the angle of reflection (r) is equal to the angle of incidence (i), and when the mirror is rotated by an angle θ, the angle of reflection changes by 2θ.
- Thus, if the angle of reflection is given by:
\[ r = \theta \]
Then, the angle through which the mirror is rotated (let’s denote it as \( \alpha \)) is:
\[ r = 2\alpha \]
Therefore, we can express the rotation angle as:
\[ \alpha = \frac{r}{2} = \frac{0.02 \text{ radians}}{2} = 0.01 \text{ radians} \]

### Step 5: Conclusion
- The angle through which the mirror is rotated is:
\[ \alpha = 0.01 \text{ radians} \]

### Final Answer:
The angle through which the mirror is rotated is \( 0.01 \text{ radians} \).

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