# Ray tracing a thin prism

1. Sep 20, 2011

### Misr

I finally found something about thin prisms on the web
I don't understand whether the angle of deviation depends on a the angle of incidence in a thin prism or not?
Is the thin prism always in the position of minimum deviation?

2. Sep 20, 2011

### xts

If you want to get understanding (rather than get ready answer to simplified yes/no question) - compute it yourself.
Compute what is deviation angle as a function of angle of incidence and apical angle, then make several plots (for several apical angles, e.g. 10°, 20°, 30°, 45°): deviation angle as a function of angle of incidence. And compare the plots.
That's pretty easy calculation. You know Snell's law. You can do it yourself! Yes, you can!

3. Sep 20, 2011

### Misr

You are right,I already tried something like that,but I failed .I always get different angles of deviation for different angles of incidence,and this should not happen,because we study that the angle of deviation in a thin prism isn't affected by the angle of incidence.

4. Sep 20, 2011

### xts

Please, post the plots you made and formulae you got - I'll try to point out what you did wrong.

5. Sep 24, 2011

### Misr

That's my work-very sorry for the bad resolution
I drew two diagrams
At A=45 degrees

-----
At A=10 degrees

The first graph looks logical but there's a math error
the second one looks very strange
I don't know any laws about thin prism ,so I drew the angles manually,and at every angle of incidence,the angle of deviation has a different value.It's not supposed to be so according to the relation
alpha=A(n-1)
so the angle of deviation is only dependent on the angle of the prism and the refractive index
Hope you could find the problem with the second graph
Thanks

Last edited: Sep 24, 2011
6. Sep 26, 2011

### Misr

I've put my work as you told me,so could you tell me what's wrong with it?

7. Sep 26, 2011

### xts

First graph looks reasonably, the second is totally wrong - check calculations (the calculator may help)

You should find first the formula describing the deflection angle as a function of incidence angle and apical angle, then make several graphs of this formula for various fixed values of apical angle.
You may want to use any computer program for drawing graphs of functions (you'd probably been taught one of such programs at school - but even excel can do this), rather than computing all that trigonometry by hand...

8. Sep 26, 2011

### Misr

which formula?I don't know how to do all of this :(

9. Sep 26, 2011

### xts

Derive a general formula, describing the deflection angle as the function of two parameters: apical angle and incidence angle.

Derivation is pretty easy. You must just draw a picture, give names (symbols, rather than actual values in degrees) to all angles, then combine and reduce several simple trigonometrical formulae.

10. Sep 26, 2011

### Misr

do u mean that
Apical angle=second Angle of incidence-Angle of refraction?
This is just for thin prism I guess.
How about math errors in both graphs

11. Sep 26, 2011

### xts

Not quite that, but that is an idea: to express the deviation angle as a function of apical and incidence angles. You must apply Snell's law twice, and express the incidence/refracted ray angles as sums of other angles.

First one seems ok (although I did not check it very carefully), the second is wrong.

12. Sep 26, 2011

### Misr

first:Is the equation I wrote correct?
Could you explain more:(
I still can't imagine what I have to do

13. Sep 26, 2011

### Misr

I checked the numbers in the second graph,there's nothing wrong with them
Could you give me some laws about thin prism to use them instead of drawing the prism manually?

14. Sep 26, 2011

### xts

There are no more laws applicable here than Snell's law and trigonometric laws for sinus of sum of angles, nor there is no need for special laws so closely related to Snell's one.
You may easily derve Misr's law: the law describing deflection angle of the prism...

first:Is the equation I wrote correct? - you haven't wrote the equation, except of sime side-notesyou used in calculations, but as you haven't presented it as a part of ordered reasoning, I can't judge them.

I still can't imagine what I have to do
Make a drawing how the ray passes through a prism. Something like in your first post. Mark all angles and give them symbols. Write down Snell's relations between incidence/refraction angles on both surfaces, using n as refraction index. Express all angles as sums/differences of: apical angle, incident angle on first surface, deflection angle.Transform those equations to have it in form:
deflection_angle = some_function_of(incident_angle, apical_angle, n)
Make graphs of that function for several fixed values of apical_angle and single example value of n
Compare those graphs visually and find some regularity of the graph shapes - it will be obvious as you look at the graphs.

Last edited: Sep 26, 2011
15. Sep 26, 2011

### Misr

you mean by "deflection angle" the angle of deviation?

16. Sep 26, 2011

### xts

Yes. Sorry for wrong terminology. The angle between final ray and incoming one.

17. Sep 26, 2011

### Misr

I've been working on this for hours
I derived two equations but there's something wrong with them
deviation angle=angle of emergence- angle of incidence+Apical angle
This works on some values of angle of incidence
but while making some calculations I found that's not true-it gives me a negative value for the angle

And it is the same for
Apical angle=Angle of refraction-second angle of incidence
Could you help me with signs because I'm getting very bored

18. Sep 27, 2011

### xts

You may chose any convention of signs you like.
I see you got lost in it. A little help from my side - I spent 10mins to make a readable drawing:

In this convention your equation is:$$\delta=\theta-\beta-\alpha$$

Please stay with naming and sign convention as on this drawing (angles are positive if they are like on the drawing, negative if they lay on opposide side of the normal to the surface), if you prefer other names and other sign convention - it is ok, but post a readable drawing.

19. Sep 28, 2011

### Misr

but according to this drawing
deviation=emergence-incidence+apical

not deviation=emergence-incidence-apical

20. Sep 28, 2011

### Misr

oh sorry you are right