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  4. How many times would a light ray be reflected inside a circular mirror?
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How many times would a light ray be reflected inside a circular mirror?

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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #20 on: 21/03/2022 12:17:57 »
Quote from: Bored chemist on 20/03/2022 22:34:13
Quote from: hamdani yusuf on 20/03/2022 22:24:44
Quote from: Bored chemist on 20/03/2022 13:11:00
Diffraction stops it being scientific.
How does a curved surface create diffraction?
I wasn't aware that anyone had said that it did.

How should I interpret this?
Quote from: Bored chemist on 20/03/2022 10:14:32
Quote from: hamdani yusuf on 20/03/2022 10:09:21
Quote from: Bored chemist on 20/03/2022 09:44:24
It would be a math problem if the mirror was a polygon.
Why does it stop being math problem when the number of sides of the polygon is increased toward infinity, i. e. becomes a  circle?
Because it stops working.
If a beam of light hits a curved surface, it's no longer a beam.
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Offline Bored chemist

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #21 on: 21/03/2022 12:27:52 »
Quote from: hamdani yusuf on 21/03/2022 12:17:57
How should I interpret this?
Carefully.

And you should interpret this



Quote from: hamdani yusuf on 20/03/2022 12:16:10
What prevents a mathematical ray from having infinitesimally narrow width? 
Quote from: Bored chemist on 20/03/2022 13:11:00
Diffraction stops it being scientific.
as saying that science- specifically diffraction- means that you can not have an infinitesimally  narrow beam of light.
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #22 on: 22/03/2022 09:20:40 »
Given the light beam is small enough compared to diameter of the mirror, and the incoming angle is selected to be a rational number of degree with small enough denominator, and the hole on the mirror surface is large enough, the light can exit from the mirror before diverging too much. In scientific education, we often find simplified problems with idealized conditions, such as negligible surface friction and air resistance, zero mass string and pulley to study classical mechanics. In electronics, we often see problems involving ideal conductors, resistors, inductors and capacitors.
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #23 on: 02/05/2022 15:55:41 »
I first encountered this problem in 1996 when I joined a scientific research competition for teenagers. A competitor came up with the question regarding reflection of a line inside a circle. By seeing his diagrams at a glance, I knew that he hadn't got the right answer.
After I got back to my hometown, I revisited the problem and finally got the answer. I also combined it with simple geometry to get general solutions for length of the lines and areas covered by those lines (like the pentagon inside a pentagram)  for any incoming angle.
I presented the problems as a prized open contest in an exhibition representing my highschool. That's when I realized that it was not a widely known math problem. At least, it wasn't a part of highschool curriculum.
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #24 on: 03/05/2022 08:24:09 »
Further exploration brought me to incomplete reflection problem, which I finally solved two years later. The problem goes like this.
A system of circular light reflector creates a pattern with s vertices and t order of reflection. How many times the light is reflected before it reaches nth point?
The key to solve it is an algorithm to calculate inverse of modulo.
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Offline Bored chemist

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #25 on: 03/05/2022 08:41:24 »
" How many times would a light ray be reflected inside a circular mirror?"
Once, after that, it's no longer a ray.
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #26 on: 05/05/2022 05:16:15 »
Quote from: Bored chemist on 03/05/2022 08:41:24
" How many times would a light ray be reflected inside a circular mirror?"
Once, after that, it's no longer a ray.
Will it stop being reflected?
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Offline Bored chemist

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #27 on: 05/05/2022 08:34:57 »
Quote from: hamdani yusuf on 05/05/2022 05:16:15
Will it stop being reflected?
In that question you use the word "it".
to what does the word refer?
You can't be referring to the light ray- because that no longer exists.
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #28 on: 06/05/2022 11:01:51 »
Quote from: Bored chemist on 05/05/2022 08:34:57
Quote from: hamdani yusuf on 05/05/2022 05:16:15
Will it stop being reflected?
In that question you use the word "it".
to what does the word refer?
You can't be referring to the light ray- because that no longer exists.
Something that doesn't exist can't be reflected.
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Offline Bored chemist

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #29 on: 06/05/2022 13:00:45 »
Quote from: hamdani yusuf on 06/05/2022 11:01:51
Something that doesn't exist can't be reflected.
So why did you ask if it was?
Quote from: hamdani yusuf on 05/05/2022 05:16:15
Will it stop being reflected?
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #30 on: 06/05/2022 23:03:50 »
Quote from: Bored chemist on 06/05/2022 13:00:45
Quote from: hamdani yusuf on 06/05/2022 11:01:51
Something that doesn't exist can't be reflected.
So why did you ask if it was?
Quote from: hamdani yusuf on 05/05/2022 05:16:15
Will it stop being reflected?

Because I don't think it would stop existing. Otherwise we would have to reject all of optics.
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Offline Bored chemist

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #31 on: 08/05/2022 10:22:01 »
Just draw what happens to a fairly narrow beam of light when it enters a hole in a hollow spherical mirror and is reflected.
How big is when it has crossed the sphere twice (i.e. through and back)?
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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #32 on: 09/05/2022 23:31:35 »
Quote from: Bored chemist on 08/05/2022 10:22:01
Just draw what happens to a fairly narrow beam of light when it enters a hole in a hollow spherical mirror and is reflected.
How big is when it has crossed the sphere twice (i.e. through and back)?
It depends on the width of the light beam and the curvature of the mirror. But you can restrict the calculations for the center of the light beam.
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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #33 on: 10/05/2022 08:38:31 »
Quote from: hamdani yusuf on 09/05/2022 23:31:35
It depends on the width of the light beam and the curvature of the mirror
OK for a beam of width W and a mirror of radius R
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Offline paul cotter

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #34 on: 10/05/2022 10:35:31 »
Caution, optics is not my area of expertise(assuming I have any!). I would imagine a narrow beam through a small hole would cause diffraction about the edge of said hole.
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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #35 on: 10/05/2022 14:20:06 »
Quote from: paul cotter on 10/05/2022 10:35:31
Caution, optics is not my area of expertise(assuming I have any!). I would imagine a narrow beam through a small hole would cause diffraction about the edge of said hole.
The hole is larger than the light beam. The edges are not shone.
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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #36 on: 10/05/2022 14:21:57 »
Quote from: Bored chemist on 10/05/2022 08:38:31
Quote from: hamdani yusuf on 09/05/2022 23:31:35
It depends on the width of the light beam and the curvature of the mirror
OK for a beam of width W and a mirror of radius R
Say W= 0.1 mm, while R=10 m
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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #37 on: 10/05/2022 18:03:45 »
Quote from: hamdani yusuf on 10/05/2022 14:21:57
Quote from: Bored chemist on 10/05/2022 08:38:31
Quote from: hamdani yusuf on 09/05/2022 23:31:35
It depends on the width of the light beam and the curvature of the mirror
OK for a beam of width W and a mirror of radius R
Say W= 0.1 mm, while R=10 m

OK, now you have taken a few days to decide what sizes you want, you can get back to what you were asked.
Quote from: Bored chemist on 08/05/2022 10:22:01
Just draw what happens to a fairly narrow beam of light when it enters a hole in a hollow spherical mirror and is reflected.
How big is when it has crossed the sphere twice (i.e. through and back)?
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Offline Bored chemist

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #38 on: 10/05/2022 18:20:40 »

* mirror ball.jpg (37.2 kB . 615x618 - viewed 964 times)Here's a very bad sketch, but it illustrates the point.
I drew the incoming beam. The thin lines represent the outside edges of the outgoing beam

If you shine a beam of light at a concave mirror it is brought to a focus.
The focal length is half the radius of curvature.
So the distance from the focal point (F) to the point (P) where the light strikes the mirror is half the radius of the sphere and, the distance to the other side of the sphere (G) is 3 times as big. (It's 3/4 times the diameter as opposed to 1/4 times the diameter)
That means the edges of the light form (roughly) two similar triangles , one 3 times as big as the other .
So the width of the beam when it strikes G is about 3 times W.

Now that (divergent) light beam is bounced back across the mirror.
If it was a parallel beam then the same thing would happen to it as happened to the original beam. It would be 3 times as big when it hit the mirror for a third time.
So it would be 9 times W.
But it was already diverging after the first reflection, so the width will be even bigger.

But to a rough approximation, the width of the beam, after n reflections is (at least) 3^n times bigger than the original beam.

This is essentially why integrating spheres work.



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Offline hamdani yusuf (OP)

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Re: How many times would a light ray be reflected inside a circular mirror?
« Reply #39 on: 11/05/2022 12:19:21 »
Quote from: Bored chemist on 10/05/2022 18:20:40
If you shine a beam of light at a concave mirror it is brought to a focus.
The focal length is half the radius of curvature.
So the distance from the focal point (F) to the point (P) where the light strikes the mirror is half the radius of the sphere and, the distance to the other side of the sphere (G) is 3 times as big. (It's 3/4 times the diameter as opposed to 1/4 times the diameter)
That means the edges of the light form (roughly) two similar triangles , one 3 times as big as the other .
So the width of the beam when it strikes G is about 3 times W.

Now that (divergent) light beam is bounced back across the mirror.
If it was a parallel beam then the same thing would happen to it as happened to the original beam. It would be 3 times as big when it hit the mirror for a third time.
So it would be 9 times W.
But it was already diverging after the first reflection, so the width will be even bigger.

But to a rough approximation, the width of the beam, after n reflections is (at least) 3^n times bigger than the original beam.
What would I get in this case?
Quote from: hamdani yusuf on 10/05/2022 14:21:57
Say W= 0.1 mm, while R=10 m
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