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Anyone know how to calculate the critical angle?

Hi there,

I have totally forgotten how to calculate the critical angle.

This is the method I have used, can anyone tell me if this is correct?

sin (i) / sin (r) = 1 / sin C

sin (i) = 53.1

sin (r) = 0.5

106.2 = 1 / sin C

sin 106.2 = 0.9602936857

1 / sin 0.9602936857

= 59.67

Is this how you work out the critical angle? Please can someone answer me, I have an exam tomorrow. Thanks. :smile:
Reply 1
Avatar for Chewwy
Chewwy
17
no. firstly, |sinz| leq 1, so sin (i) = 53.1 is rubbish. try thinking if what you say makes sense in the future.

assuming the formula in the first line to be correct:

sinI/sinR = 1/sinC
sinC = sinR/sinI
C = sin^-1 (sinR/sinI)
Ok let me just check what I think you've done there... it's a lot easier to start from first principles with these things, and the thing to start with here is Snell's law... it's all you really need to remember.

n1sinθ1=n2sinθ2n_1sin\theta_1=n_2sin\theta_2

so sinθ1sinθ2=n2n1\frac{sin\theta_1}{sin\theta_2}=\frac{n_2}{n_1}
you also have for the case of the critical angle:
n1sinθc=n2n_1sin\theta_c=n_2

so equating the two equations:

sinθ1sinθ2=sinθc\frac{sin\theta_1}{sin\theta_2}=sin\theta_c

which I believe is then the formula you need to use. To check if this makes sense one would expect the second angle to be larger since you are moving from a more to less dense medium (for TIR) and so the light moves away from the normal. This corresponds to a larger value of sin and so the equation on the left <1 as it needs to be.
Reply 3
Avatar for madima
madima
R: refractive index
C: critical angle

R = sin i / sin r
R = 1 / sin C

EDIT: this bit is just rearranging

sin i / sin r = 1 / sin C
1 / (sin i / sin r) = sin C

so C = sin^-1 (sin r / sin i)
madima
R: refractive index
C: critical angle

R = sin i / sin r
R = 1 / sin C

EDIT: this bit is just rearranging

sin i / sin r = 1 / sin C
1 / (sin i / sin r) = sin C

so C = sin^-1 (sin r / sin i)

correct me if I'm wrong here, but would that not give you a value of sin bigger than 1? To get total internal reflection you must go from a more to less dense medium, which means that the light deflects away from the normal and the angle is bigger, so sini/sinr is <1 so you'd have sin C bigger than 1, which can't happen. Therefore, somewhere you must have gone wrong. That's why it's best to go from Snell's law, it avoids mistakes.

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