Proving Fermat's principle using calculus

[christen1289]

Homework Statement

Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat's principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken.

Homework Equations

@=angle

sin(@1)/sin(@2)=v1/v2

@1= the angle of incidence
@2= the angle of refraction

The Attempt at a Solution

What is known:
1.) v1 and v2 are constants
2.) horizontal distance from A to B are constants
3.) t=d/v
v=dt
derivative of velocity= (AC)t

if the distance from the water to A is x and the distance from the water to B is y and x and y are constants then

t=csc@1x/v1 +csc@2y/v2

using this equation and the known variables i am not sure where to go from here.

 

[mighty2000]

Hello, thank you for your post. You are on the right track with using Fermat's principle to determine the path of the ray of light from point A to point B. However, there are a few things that need to be clarified in your approach.

Firstly, the distance from the water to point A and B are not constants. They will vary depending on the angle of incidence and refraction. This is because the ray of light will travel through different media at different velocities, causing it to bend and change direction. Therefore, the horizontal distance from A to B cannot be considered constant.

Secondly, your equation for time (t) is not correct. The correct equation for time is t = d/v where d is the distance traveled and v is the velocity. In this case, the distance traveled is the length of the path ACB and the velocity is the velocity of light in air (v1) or water (v2) depending on which medium the light is traveling through.

To solve this problem, you can use the principle of least time, which states that the path taken by the ray of light will be the one that minimizes the time taken. This can be expressed mathematically as follows:

t = t1 + t2

where t1 is the time taken for the light to travel from point A to the water and t2 is the time taken for the light to travel from the water to point B. Using the equation t = d/v, we can rewrite this as:

t = d1/v1 + d2/v2

where d1 is the distance traveled from point A to the water and d2 is the distance traveled from the water to point B.

To minimize this time, we can take the derivative of t with respect to the angle of incidence (@1) and set it equal to zero. This will give us the angle of incidence that minimizes the time taken for the ray of light to travel from point A to point B.

I hope this helps guide you in the right direction. Let me know if you have any further questions.