Question

Light from an argon laser strikes a diffraction grating that has 7020 slits per centimeter. The central and first-order 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 473.2 nm.

Step-by-step solution

Write down the given information

We are given the following information: - Number of slits per centimeter: 7020 - Distance between the central maximum and first-order maximum: \(0.332\,\text{m}\) - Distance from the grating to the wall: \(1.00\,\text{m}\) Our goal is to find the wavelength of the laser light.

Convert the number of slits to the grating spacing

To do this, we need to find the spacing between the slits in the diffraction grating. We are given that there are 7020 slits per centimeter. First, let's convert that value to slits per meter: \(7020\,\text{slits/cm} * \frac{100\,\text{cm}}{1\,\text{m}} = 702000\,\text{slits/m}\) Now, we can find the spacing 'd', by taking the reciprocal of the number of slits per meter: \(d = \frac{1}{702000} = 1.4251 * 10^{-6}\,\text{m}\)

Use the diffraction grating equation

The equation relating the diffraction angle '\(\theta\)', the order of the maximum 'm', the wavelength of the light '\(\lambda\)', and the grating spacing 'd' is given by: \(\sin \theta = m \frac{\lambda}{d}\) In this problem, we are finding the wavelength '\(\lambda\)' of the laser light. To do this, we'll have to find the angle '\(\theta\)' formed between the central maximum and the first-order maximum.

Calculate the angle formed

We can use the small angle approximation \(\tan \theta \approx \sin \theta \approx \theta\) for small angles. The angle '\(\theta\)' is given by the opposite side (distance between the maxima) and the adjacent side (distance from the grating to the wall): \(\theta = \frac{0.332\, \text{m}}{1.00\, \text{m}} = 0.332\, \text{rad}\)

Find the wavelength of the laser light

Now we can plug in the values into the diffraction grating equation and solve for the wavelength '\(\lambda\)': \(d = 1.4251 * 10^{-6} \,\text{m}\), \(m = 1\) (first-order maximum), and \(\theta = 0.332\, \text{rad}\). \(\sin \theta = m \frac{\lambda}{d}\) \(0.332 = 1 \frac{\lambda}{1.4251 * 10^{-6}\,\text{m}}\) Now, solve for the wavelength '\(\lambda\)': \(\lambda = 0.332 * 1.4251 * 10^{-6}\,\text{m} = 4.732 * 10^{-7}\,\text{m}\) So the wavelength of the laser light is \(4.732 * 10^{-7}\,\text{m}\) or 473.2 nm.

Key concepts

Light Wavelength

The concept of light wavelength is critical when studying diffraction gratings and laser physics. In simple terms, the wavelength of light is the distance between consecutive crests (or troughs) of a wave. This attribute of light helps us understand many of its behaviors, including how it interacts with different objects. For instance, the diffraction grating exercise involves measuring this physical property indirectly.

When light passes through a diffraction grating, which is an array of closely spaced slits, it bends and creates patterns of light and dark bands on a screen. These patterns are dependent on the wavelength of the light used and the spacing between the slits in the grating. The question presented necessitated the calculation of the wavelength based on the distance of the light bands visible on the wall. Once done, we are able to deduce that the laser emits light with a specific wavelength measured in nanometers (nm), in this case, 473.2 nm. Understanding wavelength is not only fundamental in optics but also across various scientific disciplines, ranging from quantum mechanics to telecommunications.

Laser Physics

Laser physics is a branch of optics that delves into the principles and operation of lasers. An acronym for Light Amplification by Stimulated Emission of Radiation, lasers produce coherent, monochromatic, and highly directional light. These unique characteristics stem from the process of stimulated emission where electrons in a material stimulated by an external energy source emit photons of a specific wavelength.

In the context of the exercise, we discussed an argon laser, a type of gas laser which uses argon gas as the active medium. Argon lasers are known for their ability to produce laser beams of high spectral purity at various wavelengths, one of which is precisely what we are tasked with finding out. By understanding laser physics, we can appreciate why a laser’s wavelength remains constant and can be identified by the diffraction grating pattern, a functionality imperative for various applications such as scientific research, medical procedures, and even barcode scanners.

Optics

Optics is an area of physics that is concerned with the study of light and its interactions with matter. This broad field encompasses various phenomena including reflection, refraction, diffraction, and more. Diffraction, which is the central concept in our textbook exercise, describes the bending and spreading out of waves, such as light waves, around obstacles and openings.

The experiment cited involves a diffraction grating, a tool comprised of multiple slits, to analyze the properties of light, such as its wavelength. The separation of light into its component wavelengths by a diffraction grating can be thought of as an optical spectrum analysis. By understanding how light bends and disperses when it encounters a diffraction grating, scientists and engineers can design advanced optical devices and systems. These principles of optics not only facilitate such theoretical exercises but also have practical implications in designing lenses, microscopes, and sophisticated scientific apparatus.