Apical angle in prism question

[Alteaon]

:mad:I am going through some formulas, and I'm very stuck.

The question:

What apical angle would a prism need to have if it is made of polycarbonate ( index : 1.586 ) and is to have a power of 6.

The formula I'm using to solve this -

power = 100tan angle of deviation

Another formula states that to figure out the tan angle of deviation is

tan angle of deviation= displacement in meters/distance away in meter
= P/100

Ack!
This is just an example so I have the answer, that the apical angle is 3.4 degrees. What the hell? I can't figure out how this is the solution.

six diopters would be six meters of displacement. and that divided by 100 gets .0666.... I am so undeniably confused and frustrated. I know that there has to be something here explaining this part, but I sure can't find it, and it's just irritating me more and more.
Gah!!

**This is in the Optical Formulas tutorial by Ellen Stoner, page 76

[Fezz]

:mad:

six diopters would be six meters of displacement. and that divided by 100 gets .0666.... I am so undeniably confused and frustrated.
**This is in the Optical Formulas tutorial by Ellen Stoner, page 76


Could that be wrong? Six meters of displacement?

[EyeFitWell]

My book's at home, but I remember there being a very simple formula involving the apical angle, index of refraction, and displacement. There is no angle of deviation here, you're not refracting, just displacing.

[lensgrinder]

6 = 100 tan(d)
6/100 = tan(d)
tan-1(0.06) = d
d = 3.43

Hit inv-Tan or 2nd-Tan on your calculator.

d=a(n-1)
a = apical angle
d = angle of deviation
n = refracive index

3.4 = a(0.586)
a = 3.4/0.586
a = 5.85

[Alteaon]

Thank you for your help. I was doing the problem correctly aside from not using the 2nd tan function. What is the difference between the regular tan and the 2nd tan function? I didn't read anything in the book that really explained that. It's not a conclusion I would have come up with on my own.

thank you again.

Nicole

[Alteaon]

There is another problem that states a prism is to have a power of 3.5 diopters. What deviating angle will it have...?

Don't I need to at least know what it is made from, or anything else? Is it really possible to find this out with no other information, or this a typo?

The answer given is 2.0 degrees.

I am doing fine with most other formulas, but this one particularly is vexing.

[Dave Nelson]

you need the index of refraction

[Alteaon]

Thank you all so much for your help. I am one of those people who find it hard to give up ( or take a break) from something until I have it figured it out. It's also kind of hard when some of the information is not provided. It is good to know that it's not something I can't figure out. I still don't know the difference between the 2nd function tan and the regular, default tan button on my calculator....that's just weird to me, but it will save me a lot of headaches!

[lensgrinder]

No, because your formula is:

P = 100 tan(d)
3.5/100 = tan(d)
d = tan-1(0.035)
d = 2.0

The angle of deviation is 2 degrees. The angle of deviation is equal to the apical angle, however you need to take into account your index of refraction when considering apical angle. The reason this is so is because light will slow down when it hits the prism and therefore your angle of deviation will be smaller because the light was slowed.
So let us say we need to know the apical angle if the prism was made of 1.6 with an angle of deviation of 2.0 degrees.

d= a(n-1)
2 = a (1.6-1)
2/0.6 = a
a = 3.3 degrees

[HarryChiling]

The difference between the tan functions is that when you use the tangent button it turns the angle (input) into the ratio of the opositte side divided by the adjacet side (output). When you use the inverse tangent you are turning the ratio (input) opositte divided by the adjacent into the angle (output)

tan (angle) = oposite/adjacent
angle = tan-1(oposite/adjacent)

Also as lensgrinder pointed out the deviation of light is the same no matter what index. Deviation is a matter of how much light has deviated by it's normal course, which is given to us by the 3.5 diopters. The apical angle is totally dependent on the material, because the apical angle is a definition of the actual prism which is made of the material.

[EyeFitWell]

Completely in agreement with Harry, but a simpler way to explain it is:
You know what a tangent is, right? If you type in tan(X), you get Y. But, what if you know Y and you need to know X? That's when you use Inverse tan (Y) and that gives you X, which you can check b/c tan(X)=Y

I'm sure you know, in an equation, there is always a way to "undo" each relationship. If your equation says 5X=10, you "undo" that multiplication by dividing. Inverse Sin, Cos, and Tan are all the ways to "undo" Sin, Cos, and Tan.

If your equation is Sin(x)=5, you would need the Inverse Sin of 5 to find X.

I hope that makes sense! Good luck with your studies.
(BTW, there is a Trig section in the front of your book if you get confused with the sin, tan, cos functions)

[HarryChiling]

Nice explanation, EyeFitWell :D