Lowest angle θ₁ for total internal reflection with 60° apex angle

[ParoXsitiC]

Homework Statement

Find the lowest angle of θ₁ given the apex angle is 60°. Air (n=1) is on the outside and inside (n=1.5)

ϕ is defined as 60 degrees.

Homework Equations

The Attempt at a Solution

θ_c = sin^-1($\frac{1.00}{1.50}$) = 41.81°

To my understanding, θ₁ must angle in such a way to make r (the angle of refraction) to be equal to the critical angle. At this point you will start having TIR.

They state that r = ϕ + θ_c
How? I am not seeing it.

Once I found r, I can just use snells law to find θ₁ - but I don't understand how to find r.

 

[PeterO]

Homework Statement

Find the lowest angle of θ₁ given the apex angle is 60°. Air (n=1) is on the outside and inside (n=1.5)


http://i.minus.com/1334731242/sW_NUCsWAiRgKlAAkKeqdw/ionULwjtTVYG1.png

ϕ is defined as 60 degrees.

Homework Equations

The Attempt at a Solution

θ_c = sin^-1($\frac{1.00}{1.50}$) = 41.81°

To my understanding, θ₁ must angle in such a way to make r (the angle of refraction) to be equal to the critical angle. At this point you will start having TIR.

They state that r = ϕ + θ_c
How? I am not seeing it.

Once I found r, I can just use snells law to find θ₁ - but I don't understand how to find r.

Need some sort of diagram so the position of this θ₁ is knows.

 

[ParoXsitiC]

Need some sort of diagram so the position of this θ₁ is knows.

I give one but perhaps its not showing for you since its minus.com, here it is on imgur

 

[PeterO]

I give one but perhaps its not showing for you since its minus.com, here it is on imgur

Image came through that time.

If you look at the top triangle - the one with the 60^o angle, the other two angles are (90 - r)^o and (90 - θ_c)^o

That means [(90-r) + ϕ + (90-θ_c)] = 180

so 180 -r + ϕ - θ_c = 180

or -r + ϕ - θ_c = 0 which means r = ϕ - θ_c

Notice that is slightly different to what was in your original solution. I suspect something was wrong.