Two flat rectangular mirrors, both perpendicular to a...

Two flat rectangular mirrors, both perpendicular to a horizontal sheet of paper, are set edge to edge with their reflecting surfaces perpendicular to each other. (a) A light ray in the plane of the paper strikes one of the mirrors at an arbitrary angle of incidence $\theta_{1} .$ Prove that the final direction of the ray, after reflection from both mirrors, is opposite to its initial direction. In a clothing store, such a pair of mirrors shows you an image of yourself as others see you, with no apparent right-left reversal. (b) What If? Now assume that the paper is replaced with a third flat mirror, touching edges with the other two and perpendicular to both. The set of three mirrors is called a corner-cube reflector. A ray of light is incident from any direction within the octant of space bounded by the reflecting surfaces. Argue that the ray will reflect once from each mirror and that its final direction will be opposite to its original direction. The Apollo 11 astronauts placed a panel of corner cube retro reflectors on the Moon. Analysis of timing data taken with it reveals that the radius of the Moon's orbit is increasing at the rate of $3.8 \mathrm{cm} / \mathrm{yr}$ as it loses kinetic energy because of tidal friction.

Key Concepts

Retroreflection

Retroreflection is the phenomenon where light is reflected back towards its source regardless of the angle of incidence. This occurs due to specific geometric arrangements, such as using a pair of perpendicular mirrors or a corner-cube reflector. In such systems, all reflections combine to reverse the direction of the incoming ray, making the light return parallel to the original path.

Corner-Cube Reflector

A corner-cube reflector, also known as a retroreflector, consists of three mutually perpendicular reflective surfaces. When a light ray enters the reflector, it undergoes three successive reflections, one from each surface. The unique geometry ensures that the final ray is directed back exactly opposite to its initial direction. This optical property is exploited in various applications, including surveying and space measurements.

Law of Reflection

This principle states that when a light ray strikes a reflective surface, the angle of incidence (the angle between the incoming ray and the normal to the surface) is equal to the angle of reflection (the angle between the outgoing ray and the normal). This fundamental rule is central to understanding how light behaves when it meets flat mirrors and underpins the analysis of multiple reflections.

Geometric Optics

Geometric optics is the branch of physics that deals with the propagation of light in terms of rays. It simplifies the study of light by ignoring wave effects like diffraction and interference, focusing instead on reflection and refraction phenomena. Using geometric optics, one can predict the paths of light rays as they interact with surfaces, which is essential in solving problems involving mirrors and lenses.

Transcript

00:01 So we are given the following scenario.

00:06 So we have a mirror, right? we have a mirror like this, and we have another mirror like this.

00:14 And we have a ray of light that is incident of this mirror.

00:19 So it's going to have an incident angle of 40 degrees.

00:23 Now, a law that we have learned in this chapter is that theta 1 is equal to theta prime 1.

00:30 So that essentially means that this is going to be reflected at the same angle.

00:36 So this is also going to be 40 degrees.

00:39 Now you'll notice that since this is a right angle, since these two are perpendicular to each other, this forms a right triangle, which means that this angle is going to be 90 degrees minus 40.

00:50 So this angle is going to be 90 degrees minus 40 degrees, which is going to be 50 degrees.

00:56 So this is 50 degrees.

00:58 And again, this will be reflected at an angle of 50 degrees because theta 1 is equal to theta prime of 1.

01:10 So as you can see visually, this light ray has been reflected at an angle that is opposite to the original.

01:21 And if you want to put it in the context of this angle initially, you'll see that this is, if we were to create a horizontal, and this was hitting somewhere up here, this would now be going in the opposite direction to the horizontal by 40 degrees.

01:46 So it is now indeed opposite...

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