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

Explanation:

Step1: Convert wavelength to SI unit

The wavelength of green - light $\lambda=525\ nm = 525\times10^{-9}\ m$. The formula for the position of 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 light.

Step2: Find $\theta_1$ for $m = 1$

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

Step3: Find $\theta_2$ for $m = 2$

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

Step4: Calculate angular separation

The angular separation $\Delta\theta=\theta_2-\theta_1$. Substituting the values of $\theta_1$ and $\theta_2$, we get $\Delta\theta = 18.43^{\circ}-9.06^{\circ}=9.37^{\circ}$.

Answer:

$9.37^{\circ}$