Question

A red laser pointer shines on a diffraction grating, producing a diffraction pattern on a screen behind the grating. If the red laser pointer is replaced with a green laser pointer, will the green bright spots on the screen be closer together or farther apart than the red bright spots were?

Short answer

Answer: The color of a laser affects the distance between the bright spots on a screen because the wavelength of the incident light changes the diffracted angle. In a diffraction grating, red lasers have a longer wavelength than green lasers, which results in a larger diffracted angle for red light. Therefore, the position of the bright spots for the red laser will be farther apart on the screen compared to the green laser.

Step-by-step solution

Understand diffraction grating

A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams traveling in different directions. The direction of the diffracted beams depends on the grating's spacing, incident light's wavelength, and incident angle.

Understand the formula for diffraction grating

The formula governing the diffraction pattern of diffraction grating is given by: d * sin(θ) = m * λ, where d is the grating spacing, θ is the angle of diffraction, m is the order of the diffraction, and λ is the wavelength of the incident light. Here, we want to compare the distance between the bright spots on the screen for different wavelengths.

Calculate the angles for red and green lasers

To compare the positions of the bright spots for the red laser and the green laser, we need to find the angles associated with each laser. We know that the visible light spectrum has a range of wavelengths, with red light (~700nm) having a longer wavelength compared to green light (~500nm). Since the grating spacing (d) and the display screen's distance remain constant, the angles of the diffraction will be different as we change the wavelength of the incident light. The formula for each laser can be written as: d * sin(θ_red) = m * λ_red and d * sin(θ_green) = m * λ_green.

Observe the wavelength's effect on the bright spots' distance

As the red laser has a larger wavelength than the green laser (λ_red > λ_green), we can infer from the formula that sin(θ_red) > sin(θ_green). Therefore, the diffracted angle for the red laser θ_red will be greater than the angle for the green laser θ_green.

Compare the position of bright spots for red and green lasers

As the angles of the diffracted beams are different for the red and green lasers, the location of the bright spots on the screen will also be different. Since the angle θ_red is larger than θ_green, the red bright spots will be farther apart than the green bright spots.

Key concepts

Optical Diffraction

Optical diffraction occurs when a wavefront encounters an obstacle or an opening in its path. In the context of our exercise, the diffraction grating acts as the obstacle filled with many equally spaced openings. When light from a laser pointer encounters this grating, the light bends around the openings, creating an interference pattern of bright and dark spots, known as diffraction patterns, on a screen behind the grating.

The key point to understand is that each slit within the grating causes the light wave to spread out and these spread out waves intersect with each other. Where the waves reinforce each other, you get bright spots (constructive interference), and where they cancel each other you get dark spots (destructive interference). This fascinating phenomenon allows us to precisely measure light's wavelength and other properties.

Wavelength of Light

The wavelength of light is the distance between consecutive crests or troughs in a wave and is most commonly measured in nanometers (nm), where 1 nm is one billionth of a meter. Different wavelengths of light correspond to different colors within the visible light spectrum. For instance, red light has a longer wavelength, typically around 700 nm, while green light has a wavelength near 500 nm.

When discussing the diffraction grating exercise, knowing the wavelength is crucial because it directly influences the diffraction pattern. The formula d * sin(θ) = m * λ connects the dots between the physical space on the grating, the angle at which light bends, and the wavelength of the light used, showcasing the intimate relationship between these quantities.

Angle of Diffraction

The angle of diffraction (θ) in the context of a diffraction grating is the angle at which light is scattered from its original path. It is one of the pivotal elements for interpreting the diffraction pattern on the screen. The equation d * sin(θ) = m * λ, where d is the spacing of the grating and λ is the wavelength of light, illustrates the dependence of the diffraction angle on the wavelength. This angle helps to determine the position of the bright and dark spots on the screen.

As the wavelength changes—from red to green in our exercise—the sine of the diffraction angle changes in accordance to our formula, indicating that different colors of light will diffract at different angles. This leads to the crucial insight that the distance between the bright spots will vary with the wavelength, which is integral to solving the problem at hand.

Visible Light Spectrum

The visible light spectrum comprises light of various wavelengths that are detectable by the human eye, ranging approximately from 380 nm to 740 nm. Each wavelength within this range corresponds to a different color sensation. The spectrum starts with violet light, which has the shortest wavelength observable to us, moves through the ROYGBIV colors (red, orange, yellow, green, blue, indigo, violet), and ends with red light, which has the longest wavelength.

In analyzing our exercise involving a diffraction grating, it becomes relevant that red and green light belong to different parts of this spectrum, affecting how they interact with the grating. The place of each color within the spectrum explains the observable differences in the diffraction patterns they create—specifically, why the green bright spots appear closer together compared to the red spots.