a light with a second - order dark band forms a diffraction angle of 20.0°. the diffraction grating has…

Explanation:

Step1: Calculate the grating - spacing (d)

The grating has 200.0 lines per mm. So, $d=\frac{1}{200.0}\text{ mm}=5\times 10^{-3}\text{ mm}=5\times 10^{-6}\text{ m}$.

Step2: Use the diffraction - grating formula for dark bands

The formula for dark bands in a diffraction grating is $d\sin\theta = m\lambda$, where $m$ is the order of the dark band, $\theta$ is the diffraction angle, $d$ is the grating - spacing, and $\lambda$ is the wavelength of light. Here, $m = 2$, $\theta=20.0^{\circ}$, and $d = 5\times 10^{-6}\text{ m}$. We can re - arrange the formula to solve for $\lambda$: $\lambda=\frac{d\sin\theta}{m}$. Substitute the values: $\sin\theta=\sin(20.0^{\circ})\approx0.342$, $d = 5\times 10^{-6}\text{ m}$, and $m = 2$ into the formula. $\lambda=\frac{5\times 10^{-6}\times0.342}{2}=8.55\times 10^{-7}\text{ m}$.

Step3: Convert the wavelength to nanometers

Since $1\text{ m}=10^{9}\text{ nm}$, then $\lambda=8.55\times 10^{-7}\text{ m}\times10^{9}\text{ nm/m}=855\text{ nm}$.

Answer:

855