green light (525 nm) passes through a diffraction grating with d = 3.33 x 10-6 m. what is the angular…

Explanation:

Step1: Recall the diffraction - grating formula

The formula for the maxima in a diffraction - grating is $d\sin\theta = m\lambda$, where $d$ is the grating spacing, $\theta$ is the angular position of the maximum, $m$ is the order of the maximum, and $\lambda$ is the wavelength of the light. First, convert the wavelength to SI units: $\lambda=525\ nm = 525\times10^{-9}\ m$.

Step2: Find the angle for $m = 1$

From $d\sin\theta_1=m_1\lambda$, we can solve for $\theta_1$. Given $m_1 = 1$, $d = 3.33\times10^{-6}\ m$, and $\lambda=525\times10^{-9}\ m$. Then $\sin\theta_1=\frac{m_1\lambda}{d}=\frac{1\times525\times10^{-9}}{3.33\times10^{-6}}$. $\sin\theta_1=\frac{525\times10^{-9}}{3.33\times10^{-6}}=\frac{525}{3.33\times10^{3}}\approx0.1577$. So, $\theta_1=\sin^{- 1}(0.1577)\approx9.06^{\circ}$.

Step3: Find the angle for $m = 2$

From $d\sin\theta_2=m_2\lambda$, with $m_2 = 2$, $d = 3.33\times10^{-6}\ m$, and $\lambda=525\times10^{-9}\ m$. Then $\sin\theta_2=\frac{m_2\lambda}{d}=\frac{2\times525\times10^{-9}}{3.33\times10^{-6}}$. $\sin\theta_2=\frac{1050\times10^{-9}}{3.33\times10^{-6}}=\frac{1050}{3.33\times10^{3}}\approx0.3153$. So, $\theta_2=\sin^{-1}(0.3153)\approx18.43^{\circ}$.

Step4: Calculate the angular separation

The angular separation $\Delta\theta=\theta_2-\theta_1$. $\Delta\theta = 18.43^{\circ}-9.06^{\circ}=9.37^{\circ}$.

Answer:

$9.37^{\circ}$