An optical fiber is a flexible, transparent fiber devised to transmit light between the two ends of the fiber. Complete transmission of light is achieved through total internal reflection. This problem aims to calculate the minimum index of refraction n of the optical fiber necessary to obtain total internal reflection for every possible incidence angle.
(a) Express sinθ, where the angle θ is defined in the figure above, in terms of the incidence angle α and the index of refraction n of the optical fiber. Evaluate this function for n=1.5 and α=70∘. Take the index of refraction of air to be 1.
(b) The condition on n for total internal reflection of all beams entering the fiber is achieved when θ=90∘ for all values of α. Determine the smallest value of n that satisfies that condition.
To solve this problem, we will use Snell's Law and the concept of total internal reflection.
(a) To express sinθ in terms of α and n, we can 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:
n₁ * sinα = n₂ * sinθ
where n₁ is the index of refraction of the medium the light is coming from (in this case, air) and n₂ is the index of refraction of the optical fiber.
We need to find sinθ, so we can rearrange the equation:
sinθ = (n₁/n₂) * sinα
Given that n₁ = 1 (since it's the index of refraction of air) and n₂ = 1.5, we can substitute these values into the equation:
sinθ = (1/1.5) * sinα
Now we can evaluate this function for α = 70∘:
sinθ = (1/1.5) * sin(70∘)
You can use a scientific calculator to calculate the value of sin(70∘) and then multiply it by (1/1.5) to get the result.
(b) For total internal reflection to occur, the angle of refraction (θ) must be 90 degrees (or π/2 radians). To find the minimum value of n that satisfies this condition, we need to find the maximum value of sinθ.
The maximum value of sinθ is 1, which occurs when θ = 90 degrees. So, we want to set sinθ = 1 and solve for n:
sinθ = (n₁/n) * sinα
Since sinθ = 1 and n₁ = 1, we can simplify the equation:
1 = (1/n) * sinα
Now, solve for n:
n = 1/sinα
Given that α can take any possible incidence angle, we need to find the smallest value of n that satisfies this condition. This means we need to find the maximum value of 1/sinα.
To find the maximum value of 1/sinα, we need to find the minimum value of sinα. The smallest possible value of sinα is 1, which occurs when α = 90 degrees.
So, the smallest value of n that satisfies the condition for total internal reflection for all values of α is:
n = 1/sin(90∘)
You can use a scientific calculator to calculate sin(90∘) and then take the reciprocal to get the result.