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Chapter 12: Problem 23

Suppose that \(\phi\) in Figure \(12.17\) is Brewster's angle, and that \(\theta_{1}\) is the critical angle. Find \(n_{0}\) in terms of \(n_{1}\) and \(n_{2}\).

Short Answer

Expert verified
Question: Given Brewster's angle \(\phi\), the critical angle \(\theta_{1}\), and two indices of refraction \(n_{1}\) and \(n_{2}\), find the expression for the index of refraction \(n_{0}\). Answer: The expression for the index of refraction \(n_{0}\) in terms of \(n_{1}\) and \(n_{2}\) is: \(n_{0} = \frac{n_{1}n_{2}}{\sqrt{n_{1}^{2}+n_{2}^{2}}}\)

Step by step solution

01

Write down the Snell's Law equations

By Snell's Law, we have: \(n_{0}\sin\phi = n_{1}\sin\theta_{1}\) and \(n_{1}\sin\theta_{2}=n_{2}\sin\phi\)
02

Write down the Brewster's angle condition

Brewster's angle, \(\phi\) can be calculated using: \(\tan\phi = \frac{n_{2}}{n_{1}}\)
03

Write down the relationship for the critical angle \(\theta_{1}\)

The critical angle, \(\theta_{1}\) can be calculated as: \(\sin\theta_{1}=\frac{n_{1}}{n_{0}}\)
04

Solve for \(\sin\phi\) using the Brewster's angle equation

Using the Brewster's angle equation, we can find \(\sin\phi\) as follows: \(\sin\phi = \frac{n_{2}}{\sqrt{n_{1}^{2}+n_{2}^{2}}}\)
05

Substitute \(\sin\phi\) and \(\sin\theta_{1}\) in Snell's Law equation

Now, substitute the values of \(\sin\phi\) and \(\sin\theta_{1}\) in Snell's Law equation: \(n_{0}\frac{n_{2}}{\sqrt{n_{1}^{2}+n_{2}^{2}}} = n_{1}\frac{n_{1}}{n_{0}}\)
06

Solve for \(n_{0}\)

Now we need to solve this equation for \(n_{0}\): \(n_{0}^{2} = \frac{n_{1}^{2}n_{2}^{2}}{n_{1}^{2}+n_{2}^{2}}\)
07

Write the final expression

The final expression for \(n_{0}\) in terms of \(n_{1}\) and \(n_{2}\) is: \(n_{0} = \frac{n_{1}n_{2}}{\sqrt{n_{1}^{2}+n_{2}^{2}}}\)

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