# Verifying Rparallel + Tparalllel = 1

• rogeralms
In summary, the conversation discusses using the results of Problems 4.70, which are equations (4.98) and (4.99), to show that Rparallel + Tparallel = 1. The conversation then goes on to provide the equations for Rparallel and Tparallel, and the attempt at a solution. The expert summarizer suggests using the identities ##\sin^2(x-y) = \sin^2(x+y) - \sin(2x)\sin(2y)## and ##\cos^2(x-y) = \cos^2(x+y) + \sin(2x)\sin(2y)## to solve the problem.
rogeralms

## Homework Statement

Using the results of Problems 4.70, that is EQs. (4.98) and (4.99), show that

Rparallel + Tparalllel = 1

## Homework Equations

Rparallel = ( tan^2 ( thetai - thetat) ) / (tan^2 (thetai + thetat) )

Tparallel = (sin (2*thetai) * sin (2*thetat))/ sin^2 (thetai + thetat)

## The Attempt at a Solution

After getting this far (shown below) I took it to the math help center at my university and they couldn't solve it any further than what I had done:

First put both in the same denominator

sin^2 (thetai - thetat)) / cos^2(thetai - thetat) * cos^2(thetai + thetat/sin^2(thetai + thetat which gives a common denominator of cos^2(thetai-thetat)* sin^2(thetai + thetat)

For brevity I will call thetai = i and thetat = t

Now we have sin^2(i-t)*cos^2(i+t) + sin (2*i)*sin(2*t)/ cos^2(i-t)*sin^2(i+t)

I tried (1 - cos^2(i-t)*(1-sin^2(i+t) + sin(2*i)*sin(2*t)/ cos^2(i-t)*sin^2(i+t)

which puts the minus on cos and plus angle on sin which matches the denominator but that is as far as I got which was further than the help desk at my university.

Can someone give me a hint as to which identities I should use to work this out?

You have my undying gratitude and about a million photons of positive energy sent to you for your help!

Here's two identities that might help:

##\sin^2(x-y) = \sin^2(x+y) - \sin(2x)\sin(2y)##
##\cos^2(x-y) = \cos^2(x+y) + \sin(2x)\sin(2y)##

## What is the significance of verifying Rparallel + Tparallel = 1 in scientific research?

Verifying Rparallel + Tparallel = 1 is important because it demonstrates the principles of conservation of energy and momentum in wave propagation. This relationship is a fundamental concept in physics and is essential for understanding various phenomena such as reflection, refraction, and diffraction.

## How is Rparallel + Tparallel = 1 verified in experiments?

In experiments, Rparallel + Tparallel = 1 is verified by measuring the intensity of incident and reflected waves and calculating their respective reflection and transmission coefficients. These coefficients are then compared to ensure that their sum is equal to 1, indicating that energy and momentum are conserved in the system.

## What factors can affect the accuracy of verifying Rparallel + Tparallel = 1?

Factors that can affect the accuracy of verifying Rparallel + Tparallel = 1 include experimental errors, imperfections in the materials used, and external factors such as temperature and electromagnetic interference. It is important to control these variables as much as possible in order to obtain accurate results.

## Why is it necessary to verify Rparallel + Tparallel = 1 in multiple experiments?

Verifying Rparallel + Tparallel = 1 in multiple experiments helps to ensure the reliability and validity of the results. By repeating the experiment under different conditions or using different methods, scientists can confirm that the relationship holds true and rule out any potential errors or biases in their previous experiments.

## What are the practical applications of Rparallel + Tparallel = 1 in scientific research?

The relationship between Rparallel and Tparallel has many practical applications, such as in the design and optimization of optical devices, such as lenses and mirrors. It also plays a crucial role in understanding and predicting the behavior of electromagnetic waves in various mediums, which has applications in fields such as telecommunications and remote sensing.

Replies
5
Views
2K
Replies
5
Views
1K
Replies
8
Views
1K
Replies
2
Views
1K
Replies
6
Views
3K
Replies
21
Views
3K
Replies
6
Views
2K
Replies
17
Views
3K
Replies
1
Views
1K
Replies
23
Views
1K