Determine the relationship between θB and θA

[Richie Smash]

Homework Statement

In the figure shown, a prism ABC can be used to reflect a laser beam such as WXYZ.

Determine the relationship between θB and θA

Homework Equations

Sin(c) = 1/(refractive index)

The Attempt at a Solution

To be honest I am not sure at all, I know that for total internal reflection to occur, the angle of incidence must be greater than the critical angle which is not given, but I do see that the angle WXY is 90° and since a normal is separating them then θA must be 45 degrees, am I correct in saying this? I don't know how that angle is related to θB though.

Any help would be appreciated

 

[BvU]

Make a drawing to see what you get when $\theta_b < \theta_a$.

 

[Richie Smash]

I would get refracted ray such as this?

 

[Chandra Prayaga]

Is the beam W perpendicular to the base AC?

 

[Richie Smash]

yes it is perpendicular to base AC

 

[Richie Smash]

is anyone out there?... I'm searching for the answers...

 

[haruspex]

is anyone out there?... I'm searching for the answers...

What is the relationship between θ_A and angle AXW? What is the relationship between θ_B and that angle?

 

[BvU]

yes it is perpendicular to base AC

Anything else you didn't mention in the problem statement ?

 

[Richie Smash]

Anything else you didn't mention in the problem statement ?

Sorry for not mentioning because I thought it would be understood, but all rays entering and exiting are perpendicular, and all internal relflection totally straight, so it would be safe to assume 90 degree angles where necessary, I'm sure of that.

 

[Richie Smash]

What is the relationship between θ_A and angle AXW? What is the relationship between θ_B and that angle?

Ah yes haruspex I see, θ_A and angle AXW must add up to 90 because that is a normal that is between them, as for θ_B and angle AXW...I'm not entirely sure

 

[Chandra Prayaga]

If the incoming beam is perpendicular to the base, and the beam XY is parallel to the base, there is not much choice left. Every angle in the diagram is either 90⁰, or 45⁰. The prism itself is a 45⁰ prism.

 

[BvU]

If the incoming beam is perpendicular to the base, and the beam XY is parallel to the base, there is not much choice left. Every angle in the diagram is either 90⁰, or 45⁰. The prism itself is a 45⁰ prism.

In PF, we have a strict rule (8) not to give direct answers, only hints, comments and questions...

 

[Chandra Prayaga]

Sorry about that. I will take care in the future.

 

[Richie Smash]

So if theta A and angle AXW will add up to 90, how is angleAXW related to theta B and then from there how is theta A related to theta B?

 

[BvU]

Come on !

$\quad \theta_a + \angle {\rm AXW} = 90^\circ$

$\angle {\rm AWX} = 90^\circ$ (as you gave away in a later stage )

so what about $\quad \theta_b + \angle {\rm AXW}$ ?

 

[Richie Smash]

Here goes my attempt kind physics teachers:

θB+90+(90-θA)=180
180+θB-θA=180
θB-θA=0
θB=θA

 

[BvU]

Right you are (about that kind)

Now the nice thing about this $45^\circ$ prism is that even if $\angle AXW\ne 90^\circ$ (*), the ray is still reflected in the direction it came from: WX is parallel with YZ.

A bit of practice to show that will do you good !

Greetings from a kind 'teacher'

(*) within limits determined by the Brewster angle

 

[Richie Smash]

Oh ok, so θA and θB would have been the same even if the angle wasn't 90 degrees then? Also am I safe to assume my answer is correct?

 

[BvU]

No to the first and yes to the second.

The second even with a caveat: we haven't seen that $\angle ABC$ is exactly 90 degrees in the problem statement, have we ? Are we to assume WX // YZ from thte picture, or is it a given ? because then not only $\theta_a=\theta_b$ but even stronger $\theta_a=\theta_b=45^\circ$

And for the first: if $\theta_b = 45^\circ$ and WX not $\perp{}$ AC then of course $\theta_a\ne 45 ^\circ$ so a bit more work is at hand to show that WX//YZ

 

[Chandra Prayaga]

You should draw a number of diagrams carefully. A separate diagram for each case, for example with ∠ABC = 90⁰. and ∠ABC ≠ 90⁰. Follow the rules of reflection and refraction in each case.

 

[Richie Smash]

In these questions for this level I am very sure we have to assume it's 90 degrees but I see what you are saying.