How to prove this equation mathematically (light / optics)

[Daisuke]

Homework Statement

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

Homework 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

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 into sinθa all i got was 1 = 1

 

[rock.freak667]

Homework Statement

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

Homework 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

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 into sinθa all i got was 1 = 1

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

But at θ_c, θ_a= 90 degrees. Which is the definition of the critical angle.

 

[Daisuke]

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

But at θ_c, θ_a= 90 degrees. Which is the definition of the critical angle.

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

 

[rock.freak667]

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

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 θ_a=90.

 

[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

 

[rock.freak667]

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

When going from a more optically dense medium to a less optically dense medium, it is possible for the light ray to become internally reflected (e.g. from glass to air)

(from a textbook paraphrased)

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