The index of refraction of the core of a typical fiber optic is ncore = 1.46; the cladding has nclad = 1.4. Calculate the critical angles for the total internal reflection i crit and α crit.
I have verified the answer for i crit is 74, but I cannot for the life of me figure out what α crit is. I've tried using the formula:
sin α crit = √(n^2core - n^2clad)
sin α crit = .4142463035
then α crit = sin^-1 = 24.5 degrees
but 24.5 is not the correct answer...
i crit = 74 degrees
α crit = ?
Can someone please help me out? What's the correct answer and what formula am I supposed to be using?
Thank you!
Ah, the world of fiber optics and total internal reflection, a fascinating topic indeed. Let me help you out with this conundrum.
To calculate the critical angle for total internal reflection, you are correct in using the formula:
sin α crit = √(n^2core - n^2clad)
However, it seems like there might be a small oversight in your calculation. Let's check it together:
Plugging in the values:
sin α crit = √(1.46^2 - 1.4^2)
sin α crit = √(2.1316 - 1.96)
sin α crit = √0.1716
sin α crit ≈ 0.414
Now, to find the angle α crit, we need to take the inverse sine (sin^-1) of 0.414. Let's do that:
α crit = sin^-1(0.414)
α crit ≈ 24.24 degrees
So, it appears that the correct value for α crit is approximately 24.24 degrees, not 24.5 degrees as you initially calculated.
I hope this clears up any confusion, but remember, even in the world of physics, numbers can be tricky sometimes!
To calculate the critical angle for total internal reflection in a fiber optic, you can use the following formula:
sin α crit = √(n^2core - n^2clad)
Given that the index of refraction of the core (ncore) is 1.46 and the index of refraction of the cladding (nclad) is 1.4, we can substitute these values into the formula:
sin α crit = √((1.46)^2 - (1.4)^2)
sin α crit = √(2.1316 - 1.96)
sin α crit = √0.1716
sin α crit ≈ 0.4142
Now, to find the angle α crit, you need to take the inverse sine (sin^-1) of 0.4142:
α crit ≈ sin^-1(0.4142)
α crit ≈ 24.233 degrees
The correct answer for α crit is approximately 24.233 degrees.
To calculate the critical angle α crit for total internal reflection in a fiber optic, you need to use Snell's law. Snell's law states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the indices of refraction of the two materials.
In this case, the formula is:
sin α crit = nclad / ncore
Let's plug in the values for the indices of refraction:
sin α crit = 1.4 / 1.46
Now, we need to find the inverse sine of this value to get the angle:
α crit = sin^(-1)(1.4 / 1.46)
When we calculate this, we get α crit ≈ 18.7 degrees.
So the correct answer for α crit is approximately 18.7 degrees, not 24.5 degrees.
To recap, the correct formula to use is:
sin α crit = nclad / ncore
And the correct answer for α crit is approximately 18.7 degrees, not 24.5 degrees.
I hope this clarifies things for you! Let me know if you have any further questions.