Find the first-order angular separation between wavelengths 656.3 nm and 486.1 nm using a grating with 6000 slits per cm.

Steps to Solve

1. Calculate the grating spacing

Given that there are 6000 slits per cm, the grating spacing $d$ (the distance between adjacent slits) can be calculated using: $$ d = \frac{1 \text{ cm}}{6000 \text{ slits}} = \frac{0.01 \text{ m}}{6000} = 1.67 \times 10^{-6} \text{ m} $$

2. Use the diffraction equation

For the first-order maximum (where $m = 1$), the diffraction equation is given by: $$ d \sin \theta = m \lambda $$

We need to calculate $\theta$ for both wavelengths, starting with Hα: $$ \sin \theta_{H\alpha} = \frac{m \lambda_{H\alpha}}{d} = \frac{1 \cdot 656.3 \times 10^{-9} \text{ m}}{1.67 \times 10^{-6} \text{ m}} $$ Calculating gives: $$ \theta_{H\alpha} = \arcsin\left(\frac{656.3 \times 10^{-9}}{1.67 \times 10^{-6}}\right) $$

Then the same for Hβ: $$ \sin \theta_{H\beta} = \frac{m \lambda_{H\beta}}{d} = \frac{1 \cdot 486.1 \times 10^{-9} \text{ m}}{1.67 \times 10^{-6} \text{ m}} $$ Calculating gives: $$ \theta_{H\beta} = \arcsin\left(\frac{486.1 \times 10^{-9}}{1.67 \times 10^{-6}}\right) $$

3. Calculate the angles

Using a calculator for both $\theta_{H\alpha}$ and $\theta_{H\beta}$:

• Calculate $\theta_{H\alpha}$: $$ \theta_{H\alpha} \approx 0.392 \text{ rad} \text{ (or } 22.5^\circ \text{)} $$
• Calculate $\theta_{H\beta}$: $$ \theta_{H\beta} \approx 0.291 \text{ rad} \text{ (or } 16.7^\circ \text{)} $$

4. Find the angular separation

Finally, the angular separation $\Delta\theta$ between Hα and Hβ is given by: $$ \Delta\theta = \theta_{H\alpha} - \theta_{H\beta} $$ Calculating gives: $$ \Delta\theta \approx 0.392 - 0.291 \approx 0.101 \text{ rad} \text{ (or } 5.8^\circ \text{)} $$