light with a wavelength of about 510 nm is made to pass through a diffraction grating. the angle formed…

light with a wavelength of about 510 nm is made to pass through a diffraction grating. the angle formed between the path of the incident light and the diffracted light is 12.0° and forms a first - order bright band. the number of lines per mm in the diffraction grating, rounded to the nearest whole number, is lines per mm.

light with a wavelength of about 510 nm is made to pass through a diffraction grating. the angle formed between the path of the incident light and the diffracted light is 12.0° and forms a first - order bright band. the number of lines per mm in the diffraction grating, rounded to the nearest whole number, is lines per mm.

Answer

Explanation:

Step1: Recall the diffraction - grating formula

The formula for the diffraction grating is $d\sin\theta = m\lambda$, where $d$ is the distance between the slits, $\theta$ is the angle of diffraction, $m$ is the order of the bright band, and $\lambda$ is the wavelength of the light. We are given that $\lambda = 510\ nm=510\times10^{- 9}\ m$, $\theta = 12.0^{\circ}$, and $m = 1$. We need to find $d$ first. Rearranging the formula for $d$ gives $d=\frac{m\lambda}{\sin\theta}$. Substituting the values: [d=\frac{1\times510\times 10^{-9}}{\sin(12.0^{\circ})}] Since $\sin(12.0^{\circ})\approx0.2079$, then $d=\frac{510\times 10^{-9}}{0.2079}\approx2.45\times10^{-6}\ m$.

Step2: Convert $d$ to lines per mm

The number of lines per unit length $N$ is the reciprocal of the distance between the slits $d$. First, convert $d$ to mm: $d = 2.45\times10^{-3}\ mm$. The number of lines per mm $N=\frac{1}{d}$. [N=\frac{1}{2.45\times 10^{-3}}] [N\approx408]

Answer:

408