Question

Visible light passes through a diffraction grating that has 900 slits/cm, and the interference pattern is observed on a screen that is 2.50 m from the grating. (a) Is the angular position of the first-order spectrum small enough for sin $\theta \approx \theta$ to be a good approximation? (b) In the first- order spectrum, the maxima for two different wavelengths are separated on the screen by 3.00 mm. What is the difference in these wavelengths?

Short answer

(a) Yes, the angle is small enough. (b) The wavelength difference is around 20 nm.

Step-by-step solution

Understand the approximation condition

For the approximation $\sin \theta \approx \theta$ (in radians) to be valid, $\theta$ should be small, typically less than 10 degrees. First, calculate the angle $\theta$ for the first-order spectrum to check the approximation condition.

Compute diffraction grating spacing

Convert the given grating density of 900 slits/cm to grating spacing $d$ in meters. $$d=\frac{1}{900\text{slits/cm}}=\frac{1}{90000\text{slits/m}}$$

Determine the angle $\theta$ using the first-order approximation

The angle $\theta$ for the first-order maximum is given by the formula $$sin\theta =\frac{m\lambda }{d}$$Since we are interested in the value of $\theta$ for small approximation validity, note that without a specific wavelength $\lambda$, we cannot directly calculate $\theta$ but will instead reason about the typical values in the visible light spectrum (around 500 nm) to do a comparative analysis.

Calculate angular separation for two wavelengths

Calculate the angular separation $\mathrm{\Delta }\theta$ between two wavelengths using $$\mathrm{\Delta }\theta =\frac{\mathrm{\Delta }y}{L}$$where $\mathrm{\Delta }y=3.00\text{mm}=0.003\text{m}$ (given separation on screen) and $L=2.5\text{m}$. $$\mathrm{\Delta }\theta =\frac{0.003}{2.5}$$

Compute wavelength difference

Use the relation $$\mathrm{\Delta }\theta =\frac{\mathrm{\Delta }\lambda }{d}$$Rearrange to find $$\mathrm{\Delta }\lambda =d\cdot \mathrm{\Delta }\theta$$Substitute the values of $d$ from Step 2 and $\mathrm{\Delta }\theta$ from Step 4.

Key concepts

Angular Position

Angular position in the context of a diffraction grating refers to the angle at which the different orders of maximum intensity—the bright fringes—appear on a viewing screen. This is directly related to how light waves interfere with each other after passing through the grating's slits.

To find the angular position, we use the formula $\sin \theta =\frac{m\lambda }{d}$, where $m$ is the order number, $\lambda$ is the wavelength of the light, and $d$ is the distance between grating slits. If you have visible light around wavelengths of 500 nm, to find the first-order maximum, you would use $m=1$.

For small angles, which are often the case in practical scenarios, the approximation $\sin \theta \approx \theta$ (in radians) is valid. This makes computation simpler, as you can directly measure or calculate $\theta$ without complex trigonometric functions. Checking smallness can be done by comparing $\theta$ against 10 degrees (roughly 0.1745 radians). If $\theta$ is significantly less, the approximation holds well.

Interference Pattern

When light passes through a diffraction grating, it creates an interference pattern—a series of alternating bright and dark spots or lines on a screen. This occurs due to constructive and destructive interference of light waves.

In constructive interference, the path difference between waves is a multiple of the wavelength, causing bright spots, called maxima. For destructive interference, waves cancel each other out, leading to dark spots, called minima.

The spacing between these fringes (bright or dark spots) on the screen is related to both the wavelength of the light and the distance to the screen. Interference patterns are thus directly tied to the wavelength and are a visual representation of the wave properties of light.

These patterns are noticeable and precisely measurable, making diffraction gratings powerful tools in experiments where determining light properties is essential.

Wavelength Difference

The wavelength difference between two light sources in a diffraction grating setup determines how far apart their corresponding maxima will appear on the screen. To determine the separation, one needs to consider both the distance between the grating and the screen, as well as the specific wavelengths involved.

With a known distance to the screen and a measured separation $\mathrm{\Delta }y$ on the screen between maxima, the angular separation $\mathrm{\Delta }\theta$ can be computed using $\mathrm{\Delta }\theta =\frac{\mathrm{\Delta }y}{L}$, where $L$ is the screen distance.

It's straightforward to relate this angular separation back to a difference in wavelength $\mathrm{\Delta }\lambda$ with $\mathrm{\Delta }\theta =\frac{\mathrm{\Delta }\lambda }{d}$, fully utilizing the grating spacing $d$.

The resulting difference in wavelengths is given by $\mathrm{\Delta }\lambda =d\cdot \mathrm{\Delta }\theta$, which allows determinations of wavelength differences from readily observable screen measurements—a useful method in spectroscopy.