# Non-constant index of refraction due to layered material.

1. Aug 22, 2010

### Fraqtive42

A ray of light travels through a medium with an index of refraction $$n_{1}$$ and strikes an layered medium such that the index of refraction is $$n_{2}=ky+1$$ where $$y$$ is the depth of the medium and $$k$$ is a constant. If it hits at an angle of $$\theta_{1}$$ with respect to the normal, find the angle $$\theta_{2}$$ at which the light ray refracts as a function of time.

Source: A post that I made on the Art Of Problem Solving forum.

Last edited: Aug 22, 2010
2. Aug 22, 2010

### Integral

Staff Emeritus
As with all homework like questions you must show some work before getting help.

3. Aug 22, 2010

### Fraqtive42

My work:
So far I know that $$v=\frac{c}{n_{2}}$$ is the speed of the light beam, which is also equal to $$v=\frac{dy}{dt}$$. So a differential equation to solve would be $$\frac{dy}{dt}=\frac{c}{n_{2}}$$

4. Aug 23, 2010

### ehild

The light ray does not travel along y but at an angle θ2 with respect to it. θ2 itself is a function of y.

ehild

5. Aug 23, 2010

### Swap

APhO 2004 problem 2. It is similar to this one. Look at the solution there.

6. Aug 23, 2010

### Fraqtive42

But because $$y$$ is a function of time, that also makes $$\theta_{2}$$ a function of time.

7. Aug 23, 2010

### ehild

And how are y and θ2 related?

ehild

8. Aug 23, 2010

### Fraqtive42

If the material is layered infintesimally so that the index of refraction is proportional to the y, which I stated in the problem, then y is related to $$\theta$$2 because the index of refraction is related to $$\theta$$2

9. Aug 23, 2010

### ehild

What is the relation between the refractive index and θ2?

ehild

10. Aug 23, 2010

### Fraqtive42

The refractive index and $$\theta$$2 are related through Snell's Law.

11. Aug 24, 2010

### ehild

Well. At depth y, the light ray encloses the angle θ2(y) with the y axis. The light travels along a curved path s and ds/dt = c/n2(y). At depth y, θ2 is obtained from Snell's law. Now you can set up the differential equation for θ2 as function of t.

ehild