what is the approximate wavelength of a light whose second - order dark band forms a diffraction angle of…

what is the approximate wavelength of a light whose second - order dark band forms a diffraction angle of 15.0° when it passes through a diffraction grating that has 250.0 lines per mm?\n26 nm\n32 nm\n414 nm\n518 nm

what is the approximate wavelength of a light whose second - order dark band forms a diffraction angle of 15.0° when it passes through a diffraction grating that has 250.0 lines per mm?\n26 nm\n32 nm\n414 nm\n518 nm

Answer

Explanation:

Step1: Calculate the grating - spacing (d)

The number of lines per unit length (N) is 250.0 lines/mm. The grating - spacing $d=\frac{1}{N}$. First, convert N to lines/m: $N = 250\times10^{3}$ lines/m. Then $d=\frac{1}{250\times10^{3}}m = 4\times10^{-6}m$.

Step2: Use the formula for the dark - bands in a diffraction grating

The formula for the dark - bands in a diffraction grating is $d\sin\theta=(m + 0.5)\lambda$, where m is the order of the dark - band, $\theta$ is the diffraction angle, and $\lambda$ is the wavelength of the light. Here, m = 2 and $\theta=15.0^{\circ}$. We can re - arrange the formula to solve for $\lambda$: $\lambda=\frac{d\sin\theta}{m + 0.5}$.

Step3: Substitute the values into the formula

Substitute $d = 4\times10^{-6}m$, $\theta = 15.0^{\circ}$ (so $\sin\theta=\sin(15^{\circ})\approx0.259$), and m = 2 into the formula: [ \begin{align*} \lambda&=\frac{4\times10^{-6}\times0.259}{2 + 0.5}\ &=\frac{4\times10^{-6}\times0.259}{2.5}\ &=4.144\times10^{-7}m \end{align*} ] Since $1nm = 10^{-9}m$, then $\lambda=414.4nm\approx414nm$.

Answer:

414 nm