Question

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 θ, the spot of the light is found to move through a distance y on the scale. The angle θ is given by

• A. \(\frac {y}{2x}\)
• B. \(\frac {y}{x}\)
• C. \(\frac {x}{2y}\)
• D. \(\frac {x}{y}\)

Answer 1

The Correct Option is A

To solve this problem, we need to understand the behavior of light when it reflects off a mirror that is rotated by a small angle. This involves applying the principles of reflection and geometry. Let's analyze the situation step-by-step:

1. The beam of light from the source \( L \) hits the plane mirror normally, meaning the angle of incidence is \( 0^\circ \). It gets reflected back to a spot on the scale above the source.
2. When the mirror is rotated through a small angle \( \theta \), the normal to the mirror also rotates by \( \theta \).
3. Due to the rotation of the mirror, the incident angle becomes \( \theta \) and by the law of reflection, the reflected angle also becomes \( \theta \).
4. The total change in the direction of the reflected beam (from the initial position) is \( 2\theta \). This is because a rotation of the mirror by \( \theta \) causes the already reflected incident angle to shift by an additional \( \theta \), summing up to \( 2\theta \).
5. The spot of light on the scale moves by a distance \( y \). Using simple trigonometry, for small angles, the arc length (distance \( y \)) can be approximated using \( y = 2x \tan(2\theta) \), where \( x \) is the distance to the mirror.
6. For small angles, \( \tan(2\theta) \approx 2\theta \), hence substituting, we have \( y = 2x \cdot 2\theta = 4x \theta \).
7. Solving for \( \theta\), we rearrange the formula: \( \theta = \frac{y}{4x} = \frac{y}{2x} \).

This derivation shows that the angle \( \theta \) can be expressed as:

\(\theta = \frac{y}{2x}\)

Thus, the correct answer is:

\(\theta = \frac{y}{2x}\)

Now, examining the options provided:

• \(\frac{y}{2x}\) - This is the correct option as derived above.
• \(\frac{y}{x}\) - This would imply an incorrect relationship between \( y \) and \( x \).
• \(\frac{x}{2y}\) - This rearrangement does not fit the derived formula either.
• \(\frac{x}{y}\) - Again, not compatible with the relationship established.

Therefore, the correct answer is \(\frac{y}{2x}\).