# Jenkins-White Optics: Relation between Prism/Deviation Angle and Rays

• BlazenHammer
In summary, the conversation discusses a difficult trigonometric problem that the speaker attempted for over an hour but was unable to solve. They share their thought process and steps they took, including looking up a helpful trigonometric identity. Another person offers a solution involving equations and trigonometric identities. The original speaker realizes they overcomplicated the problem and the solution is actually much simpler. The conversation ends with a reminder to complete the solution involving a left hand side.
BlazenHammer
Homework Statement
Show that, for any angle of incidence on a prism, following equation is true:
and and that the right-hand side reduces to μ' at minimum deviation.
Relevant Equations
sin[(A+d)/2]/ sin(A/2) = μcos[(r1-r2)/2]/ cos[(i-e)/2]

A = Angle of Prism
d = Total deviation by Prism
i= Angle Of First incident ray
e= Angle Of Final emergent Ray Corresponding to i
where
r1= Angle Of First refracted ray
r2= Angle Of Ray incident on second surface whose refracted ray makes angle e
μ= refractive index of prism relative to air
I've tried to attempt the first part of the problem(spent over an hour on this) as second part could be easily optained with some calculus ,I asked my friend but alas nobody could conjure the solution to this dangerous trigonometric spell.
It was just pages and pages of concoction of trigonometric manipulation with no end in sight so I think it is of no use to print my attempt here

My Thought Process: 1.) Begin with lhs, expand using sum trigonometric identity and substitute μ to arrive at something reducible to rhs
=>Failed
2.)
Try to reverse engineer the rhs into lhs , do the same circus trick of substituting μ(in in trig expression) but nothing worked
=>Failed
3.) gOOGled but found nothing except out of print solution manuals available only in libraries :(

Regards,
BrazenHammer

Please explain what μ and n' stand for. I don't want to guess.

Hello Kuruman,
I've corrected the mistake n' = μ (While i wrote it I did't knew about math symbols support :'))
μ= refractive index of prism relative to air
This problem is taken from Jenkins-White Fundamental of Optics Plane Surfaces Chapter(last problem)

I found the following trig identity to be helpful $$\sin u + \sin v = 2 \sin \left( \frac{u+v}{2} \right) \cos \left( \frac{u-v}{2} \right)$$

You need to
1. Find an expression relating A, r1 and r2.
2. Find an expression relating i, e, d and A.
3. Use Snell's law at each interface.
4. Look up the trig identity ##\sin\alpha + \sin\beta=\dots##
5. Put together all of the above equations. It pops right out.

On Edit: No need to do step 4; it has been provided by @TSny.

Thank you kuruman and TSny, I kind of made a monster out of a little moth.
Never knew this would be so simple ! :)
For others who are struggling with the same problem, here is complete solution:

## \sin{r_1} + \sin{r_2} =2\sin{\frac{r_1+r_2}{2}} \cos{\frac{r_1-r_2}{2}} - |##
## \sin{r_1} + \sin{r_2} =\frac{ \sin{i} + \sin{e}}{μ} ##
##\frac{ \sin{i} + \sin{e}}{μ} = 2\sin{\frac{i+e}{2}} \cos{\frac{i-e}{2}} - || ##

Equate (1) and (2) and the answer is right there :)

BlazenHammer said:
Thank you kuruman and TSny, I kind of made a monster out of a little moth.
Never knew this would be so simple ! :)
For others who are struggling with the same problem, here is complete solution:

## \sin{r_1} + \sin{r_2} =2\sin{\frac{r_1+r_2}{2}} \cos{\frac{r_1-r_2}{2}} - |##
## \sin{r_1} + \sin{r_2} =\frac{ \sin{i} + \sin{e}}{μ} ##
##\frac{ \sin{i} + \sin{e}}{μ} = 2\sin{\frac{i+e}{2}} \cos{\frac{i-e}{2}} - || ##

Equate (1) and (2) and the answer is right there :)
Wait a minute! This solution is only half completed. You are supposed to show that $$\frac{\sin\left(\frac{d+A}{2}\right)}{\sin\left(\frac{A}{2}\right)}=\mu\frac{\cos\left(\frac{r1-r2}{2}\right)}{\cos\left(\frac{i-e}{2}\right)}.$$What about the left hand side involving ##A##?

## 1. What is the relationship between prism/deviation angle and the deviation of rays in Jenkins-White optics?

The relationship between prism/deviation angle and the deviation of rays in Jenkins-White optics is directly proportional. This means that as the prism angle increases, the deviation angle of the rays passing through the prism also increases.

## 2. How does the material of the prism affect the deviation angle in Jenkins-White optics?

The material of the prism does not have a significant effect on the deviation angle in Jenkins-White optics. The deviation angle is primarily determined by the shape and size of the prism, not the material it is made of.

## 3. Can the deviation angle be negative in Jenkins-White optics?

Yes, the deviation angle can be negative in Jenkins-White optics. This occurs when the incident ray is entering the prism at an angle greater than the critical angle, causing total internal reflection and a negative deviation angle.

## 4. How does the number of prisms in a system affect the deviation angle in Jenkins-White optics?

The number of prisms in a system has a cumulative effect on the deviation angle in Jenkins-White optics. Each individual prism adds to the overall deviation angle, resulting in a larger total deviation angle for multiple prism systems.

## 5. What is the maximum deviation angle possible in Jenkins-White optics?

The maximum deviation angle possible in Jenkins-White optics is 180 degrees. This occurs when the incident ray is entering the prism at the critical angle, causing total internal reflection and a full 180 degree deviation.