Finding Brewster's Angle via Fresnel Equation

[876d34f]

Homework Statement

The problem about this is, which Fresnel Equation am I supposed to use?

Show that the brewster's angle is

\tan(\theta)=\frac{n_{2}}{n_{1}}

but which Fresnel equation do you use

Problem: Using the correct Fresnel Equation using (plugging in) the transmitted angle \theta_{2}

Homework Equations

I figured, If I know the correct Fresnel equation, I would be able to just set my other angle equal to 90 and then I would be able to just solve for \theta and then it would yield \tan of \theta_{2} using Snell's Formula

show or rather derive the Brewster's angle

The Attempt at a Solution

 

[876d34f]

Can someone just please verify if this is the correct procedure?

so this is what I wrote on my paper

\frac{\tan(\Theta_{1}-\Theta_{2})}{\tan(\Theta_{1}+\Theta_{2})}

\Theta_{1}+\Theta_{2}=90 Degrees

Using Snell's Law then yields n_{1}\sin(\theta_{1})=n_{2}\sin(\theta_{2})

and then that yields plugging in for \Theta_{2}=90-\Theta_{1}

yields the Brewster's Angle formula by setting R_{p} equal to 0 which is the Reflection Coefficient?

and since \sin(\theta_{2})=\sin(90-\theta_{1})=\cos(\Theta_{1})

Makes it n_{1}\sin(\Theta_{1})=n_{2}\cos(\Theta_{1})

solving for the \Theta value yields

\frac{\sin(\Theta_{1})}{cos(\Theta_{1})}=\frac{n_{2}}{n_{1}}=\tan(\Theta_{B})

Which is just equal to \tan(\Theta_{B})

 

[Redbelly98]

Welcome to Physics Forums.

I didn't follow all your details, but your final answer and starting arguments (θ₁ + θ₂ = 90^o, Snell's Law) are correct.