To determine the material index, we need to use the formula for the speed of light in a medium:
n = c/v
Where: n is the material index or refractive index, c is the speed of light in a vacuum, which is approximately 3.00 x 10^8 m/s, v is the speed of light in the material.
In this case, the pulse of light takes 2.63 ns (nanoseconds) to travel a distance of 0.500 m in the material.
To find the speed of light in the material, we can use the formula:
v = d/t
Where: v is the speed of light in the material, d is the distance traveled by light in the material, which is 0.500 m, t is the time taken by light to travel that distance, which is 2.63 ns or 2.63 x 10^(-9) s.
Substituting the values into the formula, we get:
v = 0.500 m / 2.63 x 10^(-9) s
Calculating this, we find:
v ≈ 1.90 x 10^8 m/s
Now, we can calculate the material index (n) using the speed of light in the material and the speed of light in a vacuum:
n = c / v n ≈ (3.00 x 10^8 m/s) / (1.90 x 10^8 m/s) n ≈ 1.58
Therefore, the material index (refractive index) of the given material is approximately 1.58.