Find the formula for refraction angle?

[Helly123]

Homework Statement

Homework Equations

Snell's Law
sin tetha1/v1 = sin tetha2/v2
n1 = c/v1
n1 . sin tetha 1 = n2 . sin tetha 2

The Attempt at a Solution

I try to make triangle out of line PO, so that it will have correlation to tetha, since o = center, then let's say there's point C at circle circumference. then POC is isosceles triangle.
PC^2 = PO^2 + CO^2 - 2PO.CO.cos tetha
PC^2 = 2R^2 - 2R^2 cos tetha
PC^2 = 2R^2 ( 1 - cos tetha)
n value for air = 1.00 ? but I cannot find the correlation tetha to n, and especially h...?

 

[Doc Al]

but I cannot find the correlation tetha to n, and especially h...?

Extend the line OP. That gives you the normal to the surface. The angle of incidence can be related to h & r with a bit of trig.

 

[Helly123]

Extend the line OP. That gives you the normal to the surface. The angle of incidence can be related to h & r with a bit of trig.

i tried to extend OP but still don't find the correlation with r and h

 

[Helly123]

Extend the line OP. That gives you the normal to the surface. The angle of incidence can be related to h & r with a bit of trig.

Give me more clue please?

 

[haruspex]

Give me more clue please?

Adding some more labels...
Q is the point where the vertical marked h meets the horizontal through O.
S is thepoint at the top of the line marked h.
R is a point beyond P on the line through OP.
Can you relate angle POQ to h and r?
Can you relate angle POQ to angle RPS?
Can you relate angle RPS to φ?

 

[Helly123]

Adding some more labels...
Q is the point where the vertical marked h meets the horizontal through O.
S is thepoint at the top of the line marked h.
R is a point beyond P on the line through OP.
Can you relate angle POQ to h and r?
Can you relate angle POQ to angle RPS?
Can you relate angle RPS to φ?

I tried before, is it right?
i'm not sure about the normal line I set, and theory I used

 

[Doc Al]

Now you're cooking. Looks perfect!