In 1996, the Fiberoptic Link Around the Globe (FLAG) was started. It initially involves placing a 27,000 km fiber optic cable at the bottom of the Mediterranean Sea and the Indian Ocean. Suppose the index of refraction of this fiber is 1.56. If all reflections occur at the critical angle, what is the total distance a ray of light will travel along the fiber?

To calculate the total distance traveled by a ray of light along the fiber optic cable in this scenario, we first need to understand how the critical angle and total internal reflection work.

When light travels from a medium with a higher index of refraction to a medium with a lower index of refraction, it bends away from the normal (an imaginary line perpendicular to the surface). The angle at which this bending occurs is known as the critical angle.

For total internal reflection to occur, the angle at which the light hits the boundary between the two media must be greater than the critical angle. In this case, the critical angle is based on the index of refraction of the fiber optic cable, which is given as 1.56.

Now, let's calculate the critical angle using Snell's law. Snell's law is expressed as:

n₁ * sin(θ₁) = n₂ * sin(θ₂)

Where:
- n₁ is the index of refraction of the initial medium (in this case, the fiber optic cable)
- n₂ is the index of refraction of the final medium (in this case, the surrounding medium, such as water)
- θ₁ is the angle of incidence
- θ₂ is the angle of refraction

Since all reflections occur at the critical angle, θ₂ is 90 degrees (since the light ray would travel along the surface). Therefore, sin(θ₂) is equal to 1.

Let's plug in the values and solve for θ₁:

1.56 * sin(θ₁) = 1 * 1

sin(θ₁) = 1/1.56
θ₁ = sin^(-1)(1/1.56)

Using a calculator, θ₁ ≈ 40.34 degrees.

Now, we know that in one full reflection, the light travels twice the distance it would travel in air for the same incident and refracted angles. So, the distance traveled in the fiber optic cable for each reflection would be:

2 * (27,000 km) * sin(θ₁)

Finally, to find the total distance a ray of light will travel along the fiber optic cable, assuming all reflections occur at the critical angle, we need to account for the number of reflections. The number of reflections will be the total distance divided by the distance traveled in one reflection:

Total distance = (27,000 km) * π / sin(θ₁)

Using the value of θ₁ we calculated earlier, you can substitute it into the equation above to find the total distance traveled by a ray of light along the fiber optic cable.

Duplicate post. Already answered.