A beam of light from a source L is incident normally on a plane mirror fixed at a certain distance x from the source. The beam is reflected back as a spot on a scale placed just above the source L. When the mirror is rotated through a small angle `theta` the spot of the light is found to move through a distance y on the scale. The angle `theta` is given by

Options

A. `(y)/(2x)`
B. `(y)/(x)`
C. `(x)/(2y)`
D. `(x)/(y)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Setup
We have a light source \( L \) and a plane mirror placed at a distance \( x \) from the source. The light beam hits the mirror normally (at a right angle) and reflects back to a point on a scale above the source.

**Hint:** Visualize the setup by drawing a diagram with the light source, mirror, and scale.

### Step 2: Analyze the Reflection
When the mirror is rotated by a small angle \( \theta \), the angle of incidence equals the angle of reflection. Therefore, if the mirror rotates by \( \theta \), the reflected ray will change direction by \( 2\theta \).

**Hint:** Remember that the angle of reflection equals the angle of incidence.

### Step 3: Determine the Movement of the Spot
Let \( y \) be the distance that the spot of light moves on the scale when the mirror is rotated. The distance from the source \( L \) to the mirror is \( x \).

**Hint:** Think about how the angle changes affect the position of the reflected light on the scale.

### Step 4: Relate the Angles and Distances
Using the tangent function for small angles, we can relate the angle \( 2\theta \) to the distances:
\[ \tan(2\theta) \approx 2\theta \quad \text{(for small angles)} \]
From the geometry of the situation, we have:
\[ \tan(2\theta) = \frac{y}{x} \]

**Hint:** For small angles, \( \tan \) can be approximated by the angle in radians.

### Step 5: Substitute and Solve for \( \theta \)
Substituting the approximation into the equation gives us:
\[ 2\theta = \frac{y}{x} \]
Thus, we can solve for \( \theta \):
\[ \theta = \frac{y}{2x} \]

**Hint:** Make sure to isolate \( \theta \) correctly to find its value.

### Final Result
The angle \( \theta \) is given by:
\[ \theta = \frac{y}{2x} \]