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? from chapter 16, do you recall how to relate wavelength and frequency for a transverse wave (here a light wave)?

Explanation:

Step1: Convert wavelength to meters

Given $\lambda = 632.1\ nm=632.1\times10^{-9}\ m$ and $\Delta\lambda = 0.0147\ nm = 0.0147\times10^{-9}\ m$. The speed of light $c = 3\times10^{8}\ m/s$. We use the formula $f=\frac{c}{\lambda}$, and by differentiating $f$ with respect to $\lambda$ (using the quotient rule: if $y = \frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$, here $u = c$ (constant, so $u^\prime=0$) and $v=\lambda$), we get $df=-\frac{c}{\lambda^{2}}d\lambda$. The magnitude of the frequency - width $\Delta f=\frac{c}{\lambda^{2}}\Delta\lambda$.

Step2: Calculate the frequency width

Substitute $c = 3\times10^{8}\ m/s$, $\lambda = 632.1\times10^{-9}\ m$ and $\Delta\lambda = 0.0147\times10^{-9}\ m$ into the formula $\Delta f=\frac{c}{\lambda^{2}}\Delta\lambda$.

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Answer:

$1.10\times 10^{10}\ Hz$