Question

Light from an argon laser strikes a diffraction grating that has 7020 grooves per centimeter. The central and firstorder principal maxima are separated by \(0.332 \mathrm{~m}\) on a wall \(1.00 \mathrm{~m}\) from the grating. Determine the wavelength of the laser light.

Short answer

Answer: The wavelength of the laser light is approximately \(454 \mathrm{~nm}\).

Step-by-step solution

Find the Grating Separation (d)

The grating separation (d) can be determined by the formula: \(d = \frac{1}{N}\) Where N is the number of grooves per centimeter. In this case, N = 7020 grooves per centimeter. \(d = \frac{1}{7020}\) \(d = 1.42 \times 10^{-4} \mathrm{~cm}\)

Convert Grating Separation to Meters

To work with meters in the rest of the problem, convert the grating separation (d) to meters: \(d = 1.42\times10^{-4}\ \text{cm}\times\frac{1\text{ m}}{100\text{ cm}} = 1.42\times10^{-6}\ \text{m}\)

Calculate the Angle Between Central and First-Order Maxima (θ)

We are given the distance between the central and first-order maxima (y = 0.332 m) and the distance from the grating to the wall (L = 1.00 m). We can use the right triangle relationship to find the angle θ: \(\tan \theta = \frac{y}{L}\) \(\theta = \arctan{\left(\frac{0.332 \ \text{m}}{1.00 \ \text{m}}\right)}\) \(\theta \approx 18.56^{\circ}\)

Calculate the Wavelength Using the Grating Equation

Now, we can use the grating equation to find the wavelength (λ): \(m \lambda = d \sin \theta\) Where m is the order of the principal maximum (m = 1 for the first-order maximum). \(\lambda = \frac{d \sin \theta}{m}\) \(\lambda = \frac{1.42 \times 10^{-6} \ \text{m} \cdot \sin(18.56^{\circ})}{1}\) \(\lambda \approx 4.54 \times 10^{-7} \text{ m}\)

Convert Wavelength to Nanometers

To express the wavelength in nanometers, convert the result to nanometers: \(\lambda = 4.54\times10^{-7} \text{ m}\times\frac{10^9\text{ nm}}{1\text{ m}} \approx 454\ \text{nm}\) The wavelength of the laser light is approximately \(454 \mathrm{~nm}\).

Key concepts

Wavelength Calculation

Calculating the wavelength of light in a physics problem involves understanding the relationship between diffraction gratings and light behavior. A diffraction grating consists of many parallel grooves or slits that cause the light to interfere, creating a pattern of bright and dark areas known as maxima and minima.
In this exercise, knowing the groove density (7020 grooves per centimeter) allows us to find the grating spacing, denoted by \(d\). This spacing is crucial in applying the diffraction formula. To do so:

• First, calculate the grating separation \(d\) using \(d = \frac{1}{N}\).
• Convert the unit of \(d\) from centimeters to meters for consistency in calculations.

Next, the angle \(\theta\) between the central and first-order maxima needs to be determined, utilizing the formula \(\tan \theta = \frac{y}{L}\). This angle, along with the spacing \(d\) and the diffraction order \(m = 1\), is then used in the diffraction grating equation:

• \(m \lambda = d \sin \theta\)
• Rearrange to solve for the wavelength: \(\lambda = \frac{d \sin \theta}{m}\).

Finally, convert the wavelength from meters to nanometers for a standardized format.

Laser Light

Laser light is special because it is coherent and monochromatic, meaning the light waves are in phase and consist of a single wavelength. This uniqueness makes lasers ideal for diffraction experiments.
In this exercise, the laser light hitting the diffraction grating produces a diffraction pattern. The pattern includes bright and dark bands due to interference. The bright spots, known as maxima, occur at specific angles where the light waves constructively interfere.
Lasers emit light through a process called stimulated emission, where atoms or molecules emit coherent light. Understanding these properties helps grasp why lasers and grating experiments are commonly used in optical experiments and wavelength measurements. These consistent characteristics are advantageous to display predictable diffraction patterns necessary for precise calculations.

Optics

Optics is the branch of physics that studies how light behaves and interacts with various elements like mirrors, lenses, and diffraction gratings. In this topic, concepts such as reflection, refraction, and diffraction are pivotal to understanding how light changes direction and forms patterns.
Diffraction, specifically, involves the bending of light waves around the edges of an obstacle or aperture, which can only be effectively studied using the principles of wave optics. A diffraction grating is an optical component with a periodic structure that disperses light into several beams traveling in different directions. The angle and separation of these beams are essential in optical problem-solving.
Learning about optics enables students to delve into the principles governing actions like focusing light, analyzing light spectra, and understanding phenomena such as rainbows. It bridges theoretical aspects with practical applications in areas like spectroscopy, laser technology, and even eyeglass design.

Physics Problem Solving

Solving physics problems efficiently requires a structured approach that often involves identifying known values, applying relevant equations, and verifying the results. In the context of diffraction gratings, understanding each part of the problem is vital.
First, identify and convert the given values into consistent units, as seen in our conversion from grooves per centimeter to meters. Next, use geometric relationships, such as the right triangle in this problem, to find unknown variables like angles. Finally, apply the core physics formulas, such as the grating equation, to solve for the desired quantity, ensuring each step makes logical sense.
Important tips for physics problem-solving include:

• Break down the problem step-by-step and focus on one section at a time.
• Check units for consistency throughout the calculation.
• Remember to interpret the physical significance of your final result.

By practicing these skills, students can build a strong foundation for tackling various physics challenges.