Question

What is the wavelength of a Colour of frequency 4 x 10 14 Hz in a medium of refractive index 1.5 *?

Answer 1

Approximately
`5.00 * 10^(-7)\; {\rm m}` (
`500\; {\rm nm}`.)

Step-by-step explanation:

Look up the speed of light in vacuum:
`c \approx 3.00 * 10^(8)\; {\rm m\cdot s^(-1)}`.

If the speed of light in a medium is
`v`, the refractive index of that medium will be
`n = (c / v)`. For the medium in this question, it is given that
`n = 1.5`. Rearrange the equation
`n = (c / v)\!` to find
`v`:

`\begin{aligned}v &= (c)/(n) \\ &\approx \frac{3.00 * 10^(8)\; {\rm m\cdot s^(-1)}}{1.5} \\ &\approx 2.00 * 10^(8)\; {\rm m\cdot s^(-1)}\end{aligned}`.

Let
`f` denote the frequency of this wave. When the speed of this wave is
`v`, the wavelength
`\lambda` of this wave will be
`\lambda = (v / f)`. Therefore:

`\begin{aligned}\lambda &= (v)/(f) \\ &\approx \frac{2.00* 10^(8)\; {\rm m\cdot s^(-1)}}{4 * 10^(14)\; {\rm s^(-1)}} \\ &\approx 5.00 * 10^(-7)\; {\rm m} \\ &= 500\; {\rm nm}\end{aligned}`.

(Note that
`1\; {\rm Hz} = 1\; {\rm s^(-1)}`.)