Light refracting and reflecting in a drop of water to form a rainbow

[A13235378]

Homework Statement

When a ray of sunlight enters a drop of water, it undergoes multiple internal reflections accompanied by partial transmissions out of the drop. Consider an ABCDE ray that undergoes a single internal reflection before emerging from the drop (Figure). The primary rainbow is formed when the deviation θ is minimal. Show that this happens for an incidence angle θ1r such that

Relevant Equations

n1sen theta1 = n2 sen theta 2

I drew the red and green tangent lines and I found that the angles in blue are equal to theta 1. Also , as the BCD triangle is equilateral, theta 2 = 30. With this I can calculate the side of this equilateral triangle as a function of the radius R of the circumference. After that, I can't go on. My only problem is not the physical itself, but the geometry. How can I finish?

 

[BvU]

as the BCD triangle is equilateral

Oh, is it ? How do you prove that ?

The primary rainbow is formed when the deviation θ is minimal.

What do you usually have to do to find a minimum ?

 

[A13235378]

Oh, is it ? How do you prove that ?

What do you usually have to do to find a minimum ?

So I may have rushed, because I thought that by law of reflection

BCO = OCD.

So BOC = COD

This is in fact but I cannot say that this third angle that completes an entire arc is also the same.

I really rushed, I ask for forgiveness

 

[BvU]

So it is isosceles.
(if AB is closer to the horizontal axis you can see BD is not necessarily the same length as BO )

What can you do to find an expression for $\theta$ ?

 

[A13235378]

So it is isosceles.
(if AB is closer to the horizontal axis you can see BD is not necessarily the same length as BO )

What can you do to find an expression for $\theta$ ?

I don't know if it's right, but I kind of considered the prism where S is the opening. The minimum deviation is given by the formula

D = 2i - S

Where i is the incident angle. $\theta_1$

But also the angle I marked in orange is 4 $\theta_2$

So I found that:

$\theta$ = 2 $\theta_1$ + 4 $\theta_2$ - $\pi$

Am I right?

If so, how can I proceed now to arrive at the result?

 

[hutchphd]

Please don't go hunting for some formula in the book. You know the incident angle $\theta_1$ and then Snell gives you $\theta_2$. The outgoing process is similar.
That and the symmetry is sufficient to determine all the angles using high school geometry Then you need to show that the outgoing angle has a maximum.
Just write down all the angles at point B in terms of $\theta_1$ and $\theta_2$ to start.