Light from a source A is reflected by a plane mirror to a...

Light from a source $A$ is reflected by a plane mirror to a receiver at point $B$, as shown in the accompanying figure. Show that for the light to obey Fermat's principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.) (FIGURE CAN'T COPY)

Key Concepts

Geometric Reflection Techniques

These techniques involve using geometric constructions, such as reflecting the source or target across a mirror, to transform the reflection problem into one involving straight-line distances. This approach simplifies finding the path that minimizes travel time, reinforcing the law of reflection.

Fermat's Principle

This principle states that light travels between two points along the path that requires the least time. It is a fundamental concept in optics, and by applying it, one can derive various laws governing light behavior, including reflection and refraction.

Law of Reflection

This law dictates that when light reflects off a surface, the angle of incidence is equal to the angle of reflection. It is a direct consequence of Fermat's principle, ensuring that the light follows the path that takes the least time between two points.

Transcript

00:02 Okay, when a light source, so we have a light source right here, and let's just call that point a, when a light source, one of the rays hits a flat, plain mirror, part of it will be reflected back out.

00:31 And let's say we have some kind of light receiver here to receive that reflected light and we're going to call that point b.

00:40 Okay.

00:42 And what we have is what is called for matt's principle.

00:55 And he states that light travels a path that requires minimal.

01:17 And so what we want to show is that actually will equal because when the light ray from a is reflected back we have this vertical imaginary line called the normal and so this is what is called the angle of incidence and let's label him angle one, and then we have this angle of reflection.

01:56 So theta 1 is defined to be the angle of incidence, and it's always defined from that normal line, and angle 2 is defined to be the angle of reflection.

02:16 And in order for matt's principle to be true, the angle of incidence must equal the angle of reflection.

02:30 And so we want to show that.

02:34 We want to show that.

02:36 And so what we have here, if i know this distance from here to here, is some, the horizontal distance between the source and the receiver is l.

02:51 We're going to make this distance x and so this will be l minus x and this is the height of b and this is going to be the height of a and so we know then that minimal time time is going to be equal to distance over the rate at which the light is traveling and of course that rate the light is traveling is the speed of light, which we're going to designate with c.

03:27 That is the speed of light.

03:29 And so the total time that that ray will take is going to be the distance he travels.

03:39 And so he's going to be traveling this distance right here, which is a hypotenuse of a right triangle.

03:45 So that is going to be the square root of x squared.

03:51 Plus that height squared over that speed of light.

03:57 Plus now he's going to be traveling that second path, which is l minus x squared plus the height of the receiver squared over c.

04:12 Okay...

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