Two plane mirrors a inclined at a certain angle. A ray of light first incident on one of them ant an inclination of 10° with the mirror and retraces its path after 5 reflections. The angle between the mirrors

To determine the angle between two inclined plane mirrors based on the behavior of a ray of light reflecting off them, we can use some fundamental principles of optics. When a ray of light reflects off a mirror, the angle of incidence equals the angle of reflection. This property is crucial in analyzing how light behaves when it encounters multiple mirrors.

Understanding the Reflection Process

Let’s break down the situation step by step. You have two mirrors inclined at an angle, and a ray of light strikes one of the mirrors at an angle of 10°. After five reflections, the ray retraces its path. This means that the total change in direction of the ray after all reflections must equal zero, effectively bringing it back to its original path.

Calculating the Total Angle Change

Each time the light reflects off a mirror, it changes direction by twice the angle of incidence. In this case, since the angle of incidence is 10°, the angle of reflection will also be 10°. Therefore, each reflection changes the direction of the ray by:

• Change in direction per reflection = 2 × 10° = 20°

Now, if the ray reflects off the mirrors five times, the total change in direction due to the reflections is:

• Total change = 5 × 20° = 100°

Relating Reflections to Mirror Angle

For the ray to retrace its path after five reflections, the total change in direction must be a multiple of the angle between the mirrors. If we denote the angle between the mirrors as θ, then the relationship can be expressed as:

• Total change = n × θ, where n is the number of reflections between the mirrors.

In this case, since the ray reflects five times, we can set up the equation:

• 100° = 5θ

Finding the Angle Between the Mirrors

To find θ, we can rearrange the equation:

• θ = 100° / 5 = 20°

Thus, the angle between the two mirrors is 20°. This means that each time the light reflects, it effectively interacts with the angle between the mirrors, leading to the total change in direction that allows it to return to its original path after five reflections.

Summary of Key Points

• The angle of incidence equals the angle of reflection.
• Each reflection changes the direction of the ray by twice the angle of incidence.
• The total change in direction must equal a multiple of the angle between the mirrors for the ray to retrace its path.

In conclusion, the angle between the two mirrors is 20°, which allows the ray of light to reflect five times and return to its original trajectory. This example illustrates the fascinating interplay between geometry and optics in understanding light behavior. If you have any further questions or need clarification on any part of this explanation, feel free to ask!