Question

a certain laser emits light in a narrow band of wavelengths centered at 632.1 nm and with a \wavelength width\ (such as on the scale of the figure) of 0.0147 nm. what is the corresponding \frequency width\ for the emission?

Explanation:

Step1: Convert wavelength to meters

Given $\lambda_0 = 632.1\ nm=632.1\times10^{-9}\ m$ and $\Delta\lambda = 0.0147\ nm = 0.0147\times10^{-9}\ m$.

Step2: Use the formula $c = \lambda f$ (where $c = 3\times10^{8}\ m/s$) to find $f$ and $\Delta f$

Differentiating $c=\lambda f$ (since $c$ is constant, $0 = f\Delta\lambda+\lambda\Delta f$), we get $\Delta f=\frac{f\Delta\lambda}{\lambda}$. And $f=\frac{c}{\lambda}$.
First, $f=\frac{3\times 10^{8}}{632.1\times10^{-9}}\ Hz$.
Then, $\Delta f=\frac{3\times 10^{8}}{632.1\times10^{-9}}\times\frac{0.0147\times10^{-9}}{632.1\times10^{-9}}\ Hz$.
Calculate $f=\frac{3\times10^{8}}{632.1\times10^{-9}}\approx4.746\times10^{14}\ Hz$.
$\Delta f=\frac{3\times10^{8}\times0.0147\times10^{-9}}{(632.1\times10^{-9})^{2}}\ Hz$.
$\Delta f=\frac{4.41\times10^{-2}}{3.995\times10^{-13}}\ Hz\approx1.104\times10^{11}\ Hz$.

Answer:

The frequency width $\Delta f$ is approximately $1.10\times 10^{11}\ Hz$.