Derivation of the Optical Law of Reflection

[Fernando Rios]

Homework Statement

Derive the optical law of reflection. Hint: Let light go from the point A (x1, y1) to B (x2, y,2) via an arbitrary point P = (x, 0) on a mirror along the x axis. Set dt/dx = (n/c) dD/dx = 0, where D = distance APB, and show that then theta = phi.

Relevant Equations

t = nD/c

Problem Statement: Derive the optical law of reflection. Hint: Let light go from the point A (x1, y1) to B (x2, y,2) via an arbitrary point P = (x, 0) on a mirror along the x axis. Set dt/dx = (n/c) dD/dx = 0, where D = distance APB, and show that then theta = phi.
Relevant Equations: t = nD/c

I already derived the optical law of refraction with the information given. However, I want to know why dt/dx = 0. How do I know it?

 

[collinsmark]

Problem Statement: Derive the optical law of reflection. Hint: Let light go from the point A (x1, y1) to B (x2, y,2) via an arbitrary point P = (x, 0) on a mirror along the x axis. Set dt/dx = (n/c) dD/dx = 0, where D = distance APB, and show that then theta = phi.
Relevant Equations: t = nD/c

Problem Statement: Derive the optical law of reflection. Hint: Let light go from the point A (x1, y1) to B (x2, y,2) via an arbitrary point P = (x, 0) on a mirror along the x axis. Set dt/dx = (n/c) dD/dx = 0, where D = distance APB, and show that then theta = phi.
Relevant Equations: t = nD/c

I already derived the optical law of refraction with the information given. However, I want to know why dt/dx = 0. How do I know it?

Suppose you have a function [itex] D [/itex] that represents the total distance that the light will travel from point [itex] A [/itex] to [itex] B [/itex]. You may assume that [itex] D [/itex] is a function of [itex] x [/itex]. You'll have to come up with such a function before the problem is finished, but it's not necessary to know it to answer your specific question above.

Now find a relation that shows time, [itex] t, [/itex] that the light takes to traverse that distance. Make this equation as a function of [itex] D [/itex].

Now minimize [itex] t [/itex] with respect to [itex] x [/itex].

If all is well and good, that should answer your question.

 

[Fernando Rios]

Thank you for your answer.

 

[westernira]

It is really helpful for me. Thank you for your answer.