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Phy 122

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critical angle

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In the middle of problem set 6 #24 we have

We can find the specific `thetaIn that satisfies this condition.

condition for critical angle: sin(`thetaIn) * n2 / n1 = 1

We easily solve this for `thetaIn:

critical angle: `thetaIn = arcsin(n2 / n1)

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I get the general idea but one step confuses me:

sin(`thetaIn) * n2 / n1 = 1

I would get the sin(`thetaIn)isolated on one side by multiplying both sides by n1/n2

This should give us

sin(`thetaIn) = n1/n2

And then `thetaIn = arcsin(n1/n2) which won't work since n1 is greater than n2. 1.54 > 1.19

???What am I missing???

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The condition was written carelessly.

The critical angle is that for which thetaOut = 90 deg.

Snell's Law says that

sin(thetaIn) / sin(thetaOut) = n2 / n1

so that for thetaOut = 90 deg, in which case sin(thetaOut) = 1, we have

sin(thetaIn) = n2 / n1

so that

sin(thetaIn) * n1 / n2 = 1.

In other words, the condition is

sin(thetaIn) = n2 / n1, not n1 / n2.

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