Why Does Cherenkov Radiation Form a Ring in a Concave Mirror Setup?

[ideasrule]

Homework Statement

See here: http://www.jyu.fi/kastdk/olympiads/ (year 2008, theoretical question set 2, question 2)

Let us consider a beam of particles moving with velocity v>c/n, such that the angle θ is small, along a straight line IS. The beam crosses a concave spherical mirror of focal length and center C, at point S. SC makes with SI a small angle α. The particle beam creates a ring image in the focal plane of the mirror. Explain why with the help of a sketch illustrating this fact. Give the position of the center O and the radius of the ring image.

Homework Equations

none

The Attempt at a Solution

I'm puzzling over the solutions to no avail. Why is the line from C to the center of the ring parallel to the particles' trajectory? Why is the line from C to the top of the ring, and the line from C to the bottom of the ring, parallel to the Cherenkov radiation's wavefront? I feel like I must be missing something obvious...

 

[ideasrule]

Does anybody have any idea? I have to figure this out pretty soon.

 

[turin]

I could not find the question using your link. I cannot "see" the diagram from your description. It is too confusing without the diagram. Try to attach the diagram.

 

[ideasrule]

Stupid me; I forgot that "Copy Link Location" gives a direct URL. Here it is:

http://www.jyu.fi/kastdk/olympiads/2008/Theory_2_Problem.pdf

And here's the solution:

http://www.jyu.fi/kastdk/olympiads/2008/Theory_2_Solution.pdf

 

[turin]

Consider the situation in 2 dimensions, and consider all of the rays simultaneously. This is a geometry problem. Basically, it is as if the Cherenkov rays are coming in from infinite distance (because they are parallel), and so you just need to figure out where the mirror will focus them. You have two sets of rays to consider: the ones tilted upwards and the ones tilted downwards. The mirror will focus these two sets of rays to two separate points. Those two points are two opposite points on the ring.

 

[ideasrule]

Thanks! I was missing something very simple.