Proving the Brewster's angle formula tan θ = n₂/n₁

[unscientific]

Homework Statement

I have solved the first two parts, I'm having trouble with the final part.

Given the following relations, show the following relation of brewster angle:

$$tan θ = \frac {n_2}{n_1}$$

Homework Equations

The Attempt at a Solution

Using:

$$\frac {n_2}{n1} = \frac {sin θ_1}{sin θ_3}$$

Starting from:

$$sin 2θ_3 = sin 2θ_1$$
$$sin θ_3 cos θ_3 = sin θ_1 cos θ_1$$
$$\frac {sin θ_1}{sin θ_3} = \frac {cos θ_3}{cos θ_1}$$
$$\frac {n_2}{n_1} = \frac {\sqrt {1 - sin^{2}θ_3}}{cos θ_1}$$
$$\frac {n_2}{n_1} = \sqrt { \frac {1}{sin^{2}θ_1} - ( \frac {n_1}{n_2} )^{2} } tan θ_1$$

How do i show the square root term at the bottom = 1?

 

[TSny]

Starting from:

$$sin 2θ_3 = sin 2θ_1$$

Since $\theta_1$ and $\theta_3$ lie between 0 and $\frac{\pi}{2}$, $2\theta_1$ and $2\theta_3$ lie between 0 and $\pi$.

One way to solve $\sin 2θ_3 = \sin 2θ_1$ is to have $\theta_1 = \theta_3$.

But there is also another relation between $\theta_1$ and $\theta_3$ that will satisfy $\sin 2θ_3 = \sin 2θ_1$.

 

[unscientific]

since $\theta_1$ and $\theta_3$ lie between 0 and $\frac{\pi}{2}$, $2\theta_1$ and $2\theta_3$ lie between 0 and $\pi$.

One way to solve $\sin 2θ_3 = \sin 2θ_1$ is to have $\theta_1 = \theta_3$.

But there is also another relation between $\theta_1$ and $\theta_3$ that will satisfy $\sin 2θ_3 = \sin 2θ_1$.

$$θ_1 = θ_3 + 2\pi$$

Not sure if this helps at all..

 

[TSny]

$$θ_1 = θ_3 + 2\pi$$

Not sure if this helps at all..

We need to keep $\theta_1$ and $\theta_3$ less than $\frac{\pi}{2}$, so $2\theta_1$ and $2\theta_3$ must lie between 0 and $\pi$.

Sketch a graph of the sine function between 0 and $\pi$. Draw a horizontal line that intersects the graph at two different angles. How are the two angles related?

 

[unscientific]

We need to keep $\theta_1$ and $\theta_3$ less than $\frac{\pi}{2}$, so $2\theta_1$ and $2\theta_3$ must lie between 0 and $\pi$.

Sketch a graph of the sine function between 0 and $\pi$. Draw a horizontal line that intersects the graph at two different angles. How are the two angles related?

$$θ_1 + θ_3 = \frac {\pi}{2}$$

 

[TSny]

OK. Use this in Snell's law.

 

[unscientific]

OK. Use this in Snell's law.

Ha ha, the answer just pops right out!

 

[TSny]

Ha ha, the answer just pops right out!

Good.