Question

Which type of the light incident on a grating with 1000 rulings with a spacing of \(2.00 \mu \mathrm{m}\) would produce the largest number of maxima on a screen \(5.00 \mathrm{~m}\) away? \(?\) a) blue light of wavelength \(450 \mathrm{nm}\) b) green light of wavelength \(550 \mathrm{nm}\) c) yellow light of wavelength \(575 \mathrm{nm}\) d) red light of wavelength \(625 \mathrm{nm}\) e) need more information

Short answer

Answer: Blue light with a wavelength of 450 nm.

Step-by-step solution

Write down the grating equation for constructive interference

The grating equation for constructive interference is given by: $$d\sin\theta=m\lambda$$ Here, d is the spacing between the rulings, θ is the angle of diffraction, m is the order of the maximum, and λ is the wavelength of the light.

Calculate the maximum order (m) for each wavelength

We know that for maxima to be visible, \(|\sin\theta| \leq 1\). Therefore, we can find the maximum order m for each wavelength using the inequality: $$|\frac{m\lambda}{d}| \leq 1$$ Rearranging the inequality, we get: $$m \leq \frac{d}{\lambda}$$ Now, calculate the maximum order m for each wavelength: a) For blue light (\(\lambda = 450\mathrm{nm}\)): $$m \leq \frac{(2.00\mathrm{\mu m})}{(450\mathrm{nm})}$$ b) For green light (\(\lambda = 550\mathrm{nm}\)): $$m \leq \frac{(2.00\mathrm{\mu m})}{(550\mathrm{nm})}$$ c) For yellow light (\(\lambda = 575\mathrm{nm}\)): $$m \leq \frac{(2.00\mathrm{\mu m})}{(575\mathrm{nm})}$$ d) For red light (\(\lambda = 625\mathrm{nm}\)): $$m \leq \frac{(2.00\mathrm{\mu m})}{(625\mathrm{nm})}$$

Compare the calculated maximum orders and find the answer

Now, compare the calculated maximum orders (m) for each wavelength: a) Blue light: \(m \leq \frac{(2.00\mathrm{\mu m})}{(450\mathrm{nm})} = 4.44\), so the maximum order is m = 4. b) Green light: \(m \leq \frac{(2.00\mathrm{\mu m})}{(550\mathrm{nm})} = 3.64\), so the maximum order is m = 3. c) Yellow light: \(m \leq \frac{(2.00\mathrm{\mu m})}{(575\mathrm{nm})} = 3.48\), so the maximum order is m = 3. d) Red light: \(m \leq \frac{(2.00\mathrm{\mu m})}{(625\mathrm{nm})} = 3.20\), so the maximum order is m = 3. From the above calculations, we can see that blue light with a wavelength of \(450\mathrm{nm}\) (option a) would produce the largest number of maxima (m = 4) on the screen placed 5 meters away from the grating.

Key concepts

Optics and Diffraction Gratings

Optics is a branch of physics that studies the behavior and properties of light, including its interactions with matter. Diffraction gratings are optical devices that split light into several beams traveling in different directions. They consist of many equally spaced lines, which can be realized physically by ruling many narrow lines on a reflective or transparent substrate or virtually in digital devices. The principle behind a diffraction grating is its ability to cause constructive and destructive interference among the waves of light, leading to the formation of a spectrum or pattern of light intensity called a diffraction pattern.

A key characteristic of diffraction gratings is the 'grating equation,' which showcases the relationship between the angle at which light is diffracted (\theta), the distance between the grating lines (d, also known as the grating constant), and the order of the resultant maxima (m). The grating equation, typically written as \( d\text{sin}\theta = m\text{lambda} \), is essential for understanding how different wavelengths of light (\text{lambda}) constructively interfere to produce bright spots or maxima on a screen positioned at a distance.

Constructive Interference of Light

Constructive interference occurs when two or more waves meet and combine to produce a wave with a larger amplitude than the original waves. This phenomenon is central to the workings of diffraction gratings. In the context of light and optics, when waves of light meet in phase – that is, when the peak of one wave coincides with the peak of another – they reinforce each other, resulting in brighter light. This enhancement is called constructive interference.

The conditions for constructive interference can be calculated using the grating equation and are dependent on the wavelength of the light being used as well as the spacing of the grating. A clearer understanding of this concept assists in predicting and explaining the patterns produced by a light of different wavelengths passing through a grating and helps determine how many bright fringes, or maxima, will appear on a screen placed at a distance from the grating.

The Role of Wavelength in Light Diffraction

The wavelength of light (\text{lambda}) is the distance between successive crests of a wave. It is a crucial factor in determining the behavior of light as it encounters obstacles and openings, such as a diffraction grating. In our textbook problem, different colors of light, each with a unique wavelength, are to be passed through a grating to find out which one would produce the largest number of maxima. The color that allows for the highest valid order (m) based on the grating equation will thus produce the most number of maxima.

To determine this, we utilize the relationship \( m \text{less than or equal to} \frac{d}{\text{lambda}} \) derived from the grating equation. By calculating the possible value of m for each provided wavelength, we infer that shorter wavelengths allow for higher values of m, leading to more maxima. The textbook solution confirms that blue light, with its shorter wavelength, will have the most maxima since it can accommodate a higher order of maximum when diffracted through a grating with the given specifications.