blue light (450 nm) and orange light (625 nm) pass through a diffraction grating with d = 2.88 x 10-6 m…

Explanation:

Step1: Recall the diffraction - grating formula

The diffraction - grating formula is $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the diffraction angle, $m$ is the order of diffraction, and $\lambda$ is the wavelength of light. We can solve for $\theta$: $\theta=\sin^{- 1}(\frac{m\lambda}{d})$.

Step2: Calculate the diffraction angle for blue light

Given $\lambda_{blue}=450\ nm = 450\times10^{-9}\ m$, $d = 2.88\times10^{-6}\ m$, and $m = 1$. $\theta_{blue}=\sin^{-1}(\frac{m\lambda_{blue}}{d})=\sin^{-1}(\frac{1\times450\times10^{-9}}{2.88\times10^{-6}})=\sin^{-1}(0.15625)$. $\theta_{blue}\approx9^{\circ}$.

Step3: Calculate the diffraction angle for orange light

Given $\lambda_{orange}=625\ nm = 625\times10^{-9}\ m$, $d = 2.88\times10^{-6}\ m$, and $m = 1$. $\theta_{orange}=\sin^{-1}(\frac{m\lambda_{orange}}{d})=\sin^{-1}(\frac{1\times625\times10^{-9}}{2.88\times10^{-6}})=\sin^{-1}(0.21701)$. $\theta_{orange}\approx12.5^{\circ}$.

Step4: Calculate the angular separation

The angular separation $\Delta\theta=\theta_{orange}-\theta_{blue}$. $\Delta\theta = 12.5 - 9=3.5^{\circ}$.

Answer:

$3.5$