Question

A pulse of white light is sent straight down a fiber optic cable 1 km long. The refractive index for blue light is 1.639 and for red light 1.621. What time interval will there be between the two components when they reach the far end?

Answer 1

The refractive index is also a direct measure of the factor by which the speed of light is slowed down as compared to the speed of light in a vacuum, `c(vac)= 299792458 m / s`

So the blue light is slowed down to

`c(blue)=299792458/1.639=182911811 m/s`

and the red light will be slowed down to

`c(red)=299792458/1.621=184942911 m/s`

At those speeds the light will reach the end of the cable of 1 km (=1000 m) in a time that can be calculated as
time=distance/speed or `t=s/v`

`t(red)=1000/184942911=0.000005407` sec

`->t(red)= 5407` nanoseconds
(nano- meaning one-billionth)

`t(blue)=1000/182911811=0.000005467` sec

`->t(blue= 5467` nanoseconds

The difference is `5467 -5407=60` nanoseconds

Answer:
The time interval will be (at least) 60 nanoseconds

Remark
The above calculation will only hold if the light goes through the cable in a straight line, i.e. with no or few reflections. In practice, through all the reflections, the total pathway will be longer than the kilometer in the question, and the number of reflections for blue will be more than that for red. Both of these effects will make the difference (much) greater. So consider the answer as a minimal interval .