Q.The wavelength range of the visible spectrum extends from violet (400 nm) to red (750 nm). Express these wavelengths in frequencies (Hz). ($1\ nm = 10^{-9}\ m$)

The key idea is that frequency $\nu$ and wavelength $\lambda$ are related by $\nu = c / \lambda$, where $c = 3 \times 10^8\ \text{m/s}$ is the speed of light.

1. Convert the given wavelengths from nm to m:

   $\lambda_{\text{violet}} = 400\ \text{nm} = 400 \times 10^{-9}\ \text{m} = 4 \times 10^{-7}\ \text{m}$

   $\lambda_{\text{red}} = 750\ \text{nm} = 750 \times 10^{-9}\ \text{m} = 7.5 \times 10^{-7}\ \text{m}$

2. Use $\nu = c / \lambda$ for each:

   $\nu_{\text{violet}} = \dfrac{3 \times 10^8}{4 \times 10^{-7}} = 7.5 \times 10^{14}\ \text{Hz}$ …

The key idea here is that light is an electromagnetic wave, and all electromagnetic waves travel at the same speed in vacuum: $c = 3 \times 10^8\ \text{m/s}$. The relationship between wavelength $\lambda$ and frequency $f$ is the fundamental wave equation:

For light in vacuum, $v = c$, so:

This tells us that frequency and wavelength are inversely proportional. Shorter wavelength means higher frequency, and longer wavelength means lower frequency. That's why violet light (shorter wavelength) has a higher frequency than red light (longer wavelength).

Now, we need to convert the given wavelengths from nanometers to meters, because the speed of light is in meters per second. The conversion is straightforward: $1\ \text{nm} = 10^{-9}\ \text{m}$.

1. Convert the wavelengths to meters

   Violet: $\lambda_v = 400\ \text{nm} = 400 \times 10^{-9}\ \text{m} = 4.00 \times 10^{-7}\ \text{m}$

   Red: $\lambda_r = 750\ \text{nm} = 750 \times 10^{-9}\ \text{m} = 7.50 \times 10^{-7}\ \text{m}$

2. Apply the formula $f = c / \lambda$ for violet

3. Apply the formula for red $f_r = \frac{3 \times 10^8\ \text{m/s}}{7.50 \times 10^{-7}\ \text{m}} = \frac{3}{7.50} \times 10^{8+7} = 0.40 \times 10^{15} = 4.0 \times 10^{14}\ \text{Hz}$ …