Question

If a diffraction grating produces its third-order bright band at an angle of 78.4\(^\circ\) for light of wavelength 681 nm, find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.

Short answer

13727 slits/cm; first-order at 23.4°, second-order at 53.5°; no fourth order.

Step-by-step solution

Understanding the Diffraction Formula

The diffraction grating formula relates the angles at which light of a certain wavelength will produce bright bands. It is given by \( d \sin \theta = m \lambda \), where \( d \) is the distance between slits, \( \theta \) is the angle of deflection, \( m \) is the order of the bright band, and \( \lambda \) is the wavelength of light.

Calculate Slit Separation for Third-Order Band

For the third-order band, \( m = 3 \). Given \( \theta = 78.4^{\circ} \) and \( \lambda = 681 \text{ nm} = 681 \times 10^{-9} \text{ m} \), we can rearrange the formula to find \( d \): \( d = \frac{3 \times 681 \times 10^{-9}}{\sin(78.4^{\circ})} \). Calculate \( \sin(78.4^{\circ}) \), then solve for \( d \).

Convert Slit Separation to Slits per Centimeter

We have the slit separation \( d \) in meters. To find the number of slits per centimeter, first convert \( d \) to centimeters and then take the reciprocal: \( ext{slits per cm} = \frac{1}{d\text{ in cm}} \).

First-Order Bright Band Calculation

For the first-order bright band, use \( m = 1 \). Solve the equation \( d \sin \theta = 1 \times 681 \times 10^{-9} \) for \( \theta \), using the determined \( d \) value from the previous steps: \( \sin \theta = \frac{681 \times 10^{-9}}{d} \) and solve for \( \theta \).

Second-Order Bright Band Calculation

For the second-order bright band, \( m = 2 \). Again, use the equation \( d \sin \theta = 2 \times 681 \times 10^{-9} \) to solve for \( \theta \) using the same \( d \): \( \sin \theta = \frac{2 \times 681 \times 10^{-9}}{d} \) and solve for \( \theta \).

Check for Fourth-Order Bright Band

For a fourth-order bright band, set \( m = 4 \). Use the equation \( d \sin \theta = 4 \times 681 \times 10^{-9} \). Since \( \sin \theta \leq 1 \), there will be no solution if \( \frac{4 \times 681 \times 10^{-9}}{d} > 1 \). Calculate this to confirm if the fourth-order band exists.

Key concepts

Diffraction Formula

The diffraction formula is a key concept in understanding how light interacts with a diffraction grating, which is a series of closely spaced slits. When light passes through these slits, it bends or diffracts, and can produce bright and dark bands on a screen. This phenomenon is described by the equation \( d \sin \theta = m \lambda \). Here, \( d \) stands for the distance between the slits – called the slit separation. \( \theta \) is the angle at which a bright band occurs, \( m \) represents the order number of the bright band (like first-order, second-order, etc.), and \( \lambda \) is the wavelength of the light you are using.

• \( d \): Distance between slits
• \( \theta \): Angle of diffraction
• \( m \): Order of the bright band
• \( \lambda \): Wavelength of light

The formula tells us that light of different wavelengths will diffract at different angles, creating distinct patterns of light and dark bands.

Wavelength of Light

Wavelength, denoted by \( \lambda \), is a fundamental attribute of light that significantly influences how it interacts with objects such as diffraction gratings. Essentially, the wavelength is the distance between consecutive points of a wave, like peaks or troughs. In the context of visible light, wavelengths are typically measured in nanometers (nm). For our exercise, the wavelength is 681 nm.This characteristic determines not only the color of the visible light but also how it bends when passing through or around obstacles. In a diffraction grating, the wavelength helps decide the angle at which the bright bands appear. Longer wavelengths bend less sharply than shorter ones—meaning that red light (longer wavelength) diffracts less than blue light (shorter wavelength).

Bright Band Orders

In diffraction, bright bands are the results of light waves meeting in such a way that their crests coincide, a process known as constructive interference. These bright spots occur at specific angles, and each is attributed an order number, \( m \).

• First-order band: \( m = 1 \)
• Second-order band: \( m = 2 \)
• Third-order band: \( m = 3 \), and so on.

The order number indicates how many wavelengths fit into the path difference between adjacent slits as light travels to the screen. Higher orders appear at larger angles. In our problem, we calculate the angles for the first, second, and third orders. More importantly, for orders higher than three, the calculation may show that these cannot exist if the sine function exceeds 1, as physical constraints limit \( \sin \theta \leq 1 \). This means not all orders will always appear depending on the combination of the diffraction grating's properties and the light's wavelength.

Slits Per Centimeter

A crucial detail when working with diffraction gratings is knowing how many slits contribute to the diffraction process, usually expressed as slits per centimeter. This measure provides insight into the grating's density, with more slits per centimeter resulting in greater diffraction and more detailed separation of light.To find slits per centimeter, first get the slit separation \( d \) using the diffraction formula, then convert \( d \) into centimeters. The number of slits per centimeter is simply the reciprocal of this value. Larger values here mean more slits packed into a small space, enhancing the grating's ability to produce clear, distinct interference patterns. This concept allows us to understand and predict the angles at which light's bright bands will form for various orders.