Problem 39 - Physics

Question

For a wavelength of \(420 \mathrm{~nm}\), a diffraction grating produces a bright fringe at an angle of \(26^{\circ}\). For an unknown wavelength, the same grating produces a bright fringe at an angle of \(41^{\circ} .\) In both cases the bright fringes are of the same order \(m\). What is the unknown wavelength?

Step-by-Step Solution

Answer

The unknown wavelength is approximately 628 nm.

Step 1: Understand the Diffraction Grating Equation

The diffraction grating equation is given by \(d \sin \theta = m \lambda\), where \(d\) is the distance between slits in the grating, \(\theta\) is the angle of the diffracted light, \(m\) is the order of the fringe, and \(\lambda\) is the wavelength of light. In both cases provided, the order \(m\) is the same.

Step 2: Express the Equation for Known Wavelength

For the known wavelength \(\lambda_1 = 420 \text{ nm}\) and angle \(\theta_1 = 26^{\circ}\), the diffraction equation is:\[d \sin(26^{\circ}) = m \cdot 420 \, \text{nm}\]

Step 3: Express the Equation for Unknown Wavelength

For the unknown wavelength \(\lambda_2\) and angle \(\theta_2 = 41^{\circ}\), the diffraction equation is:\[d \sin(41^{\circ}) = m \cdot \lambda_2\]

Step 4: Divide the Equations to Eliminate m and d

Since the order \(m\) and the slit separation \(d\) are the same, divide the equation for the unknown wavelength by the equation for the known wavelength:\[\frac{d \sin(41^{\circ})}{d \sin(26^{\circ})} = \frac{m \cdot \lambda_2}{m \cdot 420 \, \text{nm}}\]This simplifies to:\[\frac{\sin(41^{\circ})}{\sin(26^{\circ})} = \frac{\lambda_2}{420 \, \text{nm}}\]

Step 5: Solve for Unknown Wavelength \(\lambda_2\)

Solve the simplified equation for \(\lambda_2\):\[\lambda_2 = 420 \, \text{nm} \times \frac{\sin(41^{\circ})}{\sin(26^{\circ})}\]Calculate the values:\[\lambda_2 \approx 420 \, \text{nm} \times \frac{0.6561}{0.4384} \approx 420 \, \text{nm} \times 1.4962 \approx 628.404 \, \text{nm}\]

Step 6: Conclusion

Therefore, the unknown wavelength \(\lambda_2\) is approximately 628 nm.

Key Concepts

Wavelength Calculation

Wavelength calculation is all about determining the length of light waves, which is essential in understanding how light interacts with objects. In the context of a diffraction grating, this concept becomes particularly interesting because a diffraction grating can diffract light into different directions, creating a pattern, typically seen as bright and dark bands known as "fringes." These patterns depend on the wavelength of light used.

To calculate the wavelength, we use the diffraction grating equation: \(d \sin \theta = m \lambda\), where:

• \(d\) is the spacing between the slits in the grating.
• \(\theta\) is the angle at which light is diffracted.
• \(m\) is the order number of the fringe.
• \(\lambda\) is the wavelength of the light.

By solving for \(\lambda\), we can figure out unknown wavelengths. In scenarios where you have a known wavelength and its corresponding diffraction angle, you can use these values to solve for an unknown wavelength by comparing the equations for both the known and unknown values. This comparison allows eliminating other variables like order \(m\) and slit distance \(d\).

Therefore, wavelength calculation using a diffraction grating not only helps in identifying unknown light properties but also deepens understanding of how light behaves upon interaction with obstacles.

Angle of Diffraction

The angle of diffraction is a key factor in how light spreads out or bends through a diffraction grating. This angle, denoted as \(\theta\), determines the direction in which different wavelengths of light are distributed after passing through a grating.

This happens because light waves encountering the grating will interfere with each other, resulting in new wavefronts that travel at various angles. The essence of this phenomenon is captured mathematically by the diffraction grating equation: \(d \sin \theta = m \lambda\). Here, \(\theta\) plays a significant role as it correlates with the possible observable spectrum of colors, each corresponding to a specific wavelength of light.

When doing an experiment involving diffraction, measuring \(\theta\) precisely is crucial because even a small error can significantly affect the calculated wavelength, leading to errors in determining the physical properties of light. The angle \(\theta\) changes with the order of fringe \(m\) and can increase or decrease depending on the setup and the properties of the grating. Due to these dependencies, understanding the angle of diffraction is critical in scientific research and engineering applications where exact light behavior characterization is necessary.

Order of Fringe

The order of fringe \(m\) in diffraction refers to the position of a bright or dark band in the interference pattern formed by light as it passes through a diffraction grating. Each order corresponds to a different set of conditions where constructive interference occurs.

In the diffraction grating equation \(d \sin \theta = m \lambda\), \(m\) is an integer that denotes the sequence of intensity maxima on either side of the central maximum. When \(m=1\), it refers to the first order fringe, which is typically less intense than the central maximum (zero order).

Higher orders (\(m=2, 3, 4,\) etc.) represent points along the pattern where light constructively interferes at higher angles. However, not all orders might be visible, depending on the wavelength, slit separation, and overall diffraction setup. Understanding the order of fringes is crucial because it allows scientists to calculate wavelengths accurately, determine the type of light source being utilized, and analyze any changes occurring in mediums or materials as light passes through them.