Application of the euler equations using fermats principle

[paddy543211]

Homework Statement

The speed of light in a medium with index of refraction n is c/n, where c is the speed of light in vacuum. Notice that n ≥1:
Suppose a light ray travels in the xy-plane between (x1; y1) and (x2; y2) in a non-uniform
material so that n(x) is the refractive index of the material at x.
Show that the path of the ray is y(x) = y1 + ∫(x1 -> x) z/(sqrt(n^2(x') - z^2) dx' ,where z is a constant

Homework Equations

Euler's Equations

The Attempt at a Solution

I am aware you are suppose to split up the path into strips of thickness dx but am unaware of how to approach this. Solution would be very helpful. Thanks

 

[mark726]

.

Hello,

Thank you for your post. It seems like you are trying to solve a problem related to the path of a light ray in a non-uniform medium. This problem can be solved using Euler's equations, which relate the refractive index of a medium to the path of a light ray.

To solve this problem, we can use the following steps:

1. Start by considering a small strip of thickness dx at a point x along the path of the light ray. Let the refractive index of this strip be n(x).

2. Using Euler's equations, we can write the angle of incidence at this point as θ = tan^-1(y'(x)), where y'(x) is the derivative of the path with respect to x.

3. Similarly, the angle of refraction can be written as φ = sin^-1(sin(θ)/n(x)). This is because the light ray travels from a medium with refractive index n(x) to vacuum with refractive index 1.

4. Now, using Snell's law, we can write the relationship between the angles of incidence and refraction as sin(θ)/sin(φ) = n(x).

5. Substituting the values of θ and φ from step 2 and 3, we get y'(x) = n(x)√(1-y'(x)^2).

6. Solving for y'(x), we get y'(x) = z/√(n(x)^2-z^2), where z is a constant.

7. Integrating this equation with respect to x, we get y(x) = y1 + ∫(x1 -> x) z/(sqrt(n^2(x') - z^2) dx', where y1 is the initial position of the light ray at point (x1, y1).

I hope this helps you in solving the problem. Please let me know if you have any further questions.