Physics Geniuses: Mirrors;How many images Produced?

[TipTip]

Homework Statement

In the following diagram, why not 6 images are produced? Explain thoroughly. And mention the amount of images produced.

Problem Image:

http://img691.imageshack.us/img691/8315/imageproduced.jpg

or
http://file15.9q9q.net/preview/92743229/Image-Produced.jpg.htmlPlease Answer thoroughly and if possible, please include a sketch of

the amount of images produced.

If you can solve this, then you are a REAL Genius.

Thanks.

Homework Equations

N/A

The Attempt at a Solution

Well, I tried to project the images and I got 6 images, one from each mirror. another two from the imaginary mirror of the two mirrors and another 2 from that reflected mirror. But actually our phd professor told me that it is less than 6.TipTip

 

[TipTip]

Any solution?

 

[jdwood983]

If you had one mirror, how many images would you expect?

 

[TipTip]

one mirror = one image

 

[jdwood983]

one mirror = one image

Correct. And it doesn't matter where the object is in relation to the mirror, I should still only see 1 image, right?

 

[jdwood983]

The only thing that matters for finding the number of images produced, when there are two mirrors, is the angle between the mirrors. The formula needed is given by

$$n=\frac{360^\circ}{\theta}-1$$

So in your problem, $$\theta=72^\circ$$. Using the formula:

$$n=\frac{360}{72}-1=5-1=4$$

Now does this answer make sense? There should be 1 image produced from each of the two mirrors (2 images). Then the two images each create one more image (4 images total now). I briefly searched and found a diagram showing the line tracing, but it is for mirrors at $$90^\circ$$ so it's not really the same problem:
http://electron9.phys.utk.edu/optics421/modules/m1/images/Image1309.gif

 

[TipTip]

Still. the correct answer is 5 but I don't know the reason

 

[jdwood983]

Still. the correct answer is 5 but I don't know the reason

I believe your professor is erroneous. There are 5 total objects, but 4 of them are images. I found a good java applet online to show you how and where the images are created:
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=569.0

 

[TipTip]

Actually, no, that applet is not correct, based [1], number of images produced by hinged mirrors at an angle "x" can be found using:

1. If 180°/q = x, an integer then the number of image formed, N = 2x-1, regardless of where the object is.
2. If 180°/q = x + 0.5, number of images formed , N = 2x when the object lies on the angle bisector and N = 2x+1 on other positions.
3. If 180°/q = x + n/q and n/q < 0.5 then the number of images formed, N = 2x with the location of the object anywhere on the central angular sector of (q-2n)° about the angle bisector. N = 2x+1 when the object lies outside of the given sector.
4. If 180°/q = x + n/q and n/q > 0.5 then the number of images formed, N = 2x + 2 with the location of the object anywhere on the central angular sector of (2n-q)° about the angle bisector. N = 2x + 1 when the object lies outside of the given sector. if we use equation 2, then we get 2.5= 2*2+1 = 5 images.

What I want to know, why is this so.

[1] V. M. Kulkarni. "Number of Images Produced by Multiple Reflection".American Journal of Physics Volume 28 (1960): 317-318

 

[jdwood983]

Actually, no, that applet is not correct, based [1], number of images produced by hinged mirrors at an angle "x" can be found using:

1. If 180°/q = x, an integer then the number of image formed, N = 2x-1, regardless of where the object is.
2. If 180°/q = x + 0.5, number of images formed , N = 2x when the object lies on the angle bisector and N = 2x+1 on other positions.
3. If 180°/q = x + n/q and n/q < 0.5 then the number of images formed, N = 2x with the location of the object anywhere on the central angular sector of (q-2n)° about the angle bisector. N = 2x+1 when the object lies outside of the given sector.
4. If 180°/q = x + n/q and n/q > 0.5 then the number of images formed, N = 2x + 2 with the location of the object anywhere on the central angular sector of (2n-q)° about the angle bisector. N = 2x + 1 when the object lies outside of the given sector.


if we use equation 2, then we get 2.5= 2*2+1 = 5 images.

What I want to know, why is this so.

[1] V. M. Kulkarni. "Number of Images Produced by Multiple Reflection".American Journal of Physics Volume 28 (1960): 317-318

You can actually move the object around in between the mirrors, so the applet really is correct. Moving the object closer to the side, there are 5 images produced as you anticipated.

I had never seen any of the 4 above equations before, I have only ever seen the one equation I wrote.

The reason why there are any images produced comes from line tracing, which the applet does for you. If you check the box next to 'initialize' you will see the rays traced out that show where each image comes from as well as the two 'image' mirrors that exist. But the basic idea is still line tracing.

 

[TipTip]

Well, yes you are right.
Now, I do understand how to find number of images but still I don't get the line tracing thing.
I contacted Dr Walter H.G. Lewin, Physics MIT lecturer, but unfortunately he replied that my question requires more than 10 minutes of his time and that's why he can't answer me.

 

[jdwood983]

Well, yes you are right.
Now, I do understand how to find number of images but still I don't get the line tracing thing.
I contacted Dr Walter H.G. Lewin, Physics MIT lecturer, but unfortunately he replied that my question requires more than 10 minutes of his time and that's why he can't answer me.

There are books aplenty on ray tracing techniques. Any Introduction to Optics textbook (Pedrotti, Pedrotti, Pedrotti comes to mind first) should give ample information how how ray tracing works. A pdf I found after a quick google search of 'ray tracing planar mirrors optics' might help introduce the idea in a few minutes worth of reading http://www.physnet.org/modules/pdf_modules/m260.pdf