# How to prove this equation mathematically (light / optics)

1. Jul 22, 2009

### Daisuke

1. The problem statement, all variables and given/known data

Prove mathematically that, for a material m in air, the since of the critical angle is given by the expression sinθc = 1/nm

2. Relevant equations
n = index of refraction
m = material
a = air
na = 1
critical angle = 90 degrees

nm x sinθc = na x sinθa

or

Sinθa/Sinθc = nm

3. The attempt at a solution
I thought of plugging in the variables in the first equation and i tried rearraging it but I got this
nm x sinθc = 1 x sinθa
nm x sin90 = 1 x sinθa
sinθa = nm
and if i plugged that in to sinθa all i got was 1 = 1
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 22, 2009

### rock.freak667

I'm not sure what you mean by m= material....

But at θc, θa= 90 degrees. Which is the definition of the critical angle.

3. Jul 22, 2009

### Daisuke

Well θc = 90 degrees however, I thought the incident angle won't be 90 degrees celcius because i don't think both air and the unkown material have the same index of refraction

4. Jul 22, 2009

### rock.freak667

Well I assumed the light was going from the medium to the air (more optically dense to less optically dense medium).

so the angle of refraction θa=90.

5. Jul 22, 2009

### Daisuke

No is the other way around the air into the medium I think so the angle of incident = something and angle of refraction in the medium = 90

6. Jul 22, 2009

### rock.freak667

(from a textbook paraphrased)

$$n_1 sin\theta_1 = n_2 sin\theta_2$$