Question

Two different wavelengths of light are incident on a diffraction grating. One wavelength is \(600 . \mathrm{nm}\), and the other is unknown. If the third- order bright fringe of the unknown wavelength appears at the same position as the second-order bright fringe of the \(600 .-\mathrm{nm}\) light, what is the value of the unknown wavelength?

Short answer

Answer: The value of the unknown wavelength is 400 nm.

Step-by-step solution

Apply the bright fringe formula for the \(600\) nm wavelength

We have the formula for the bright fringe in a diffraction grating as \(d \sin \theta = m \lambda\). For the known \(600\) nm light, we have the second-order bright fringe, so we set the values of \(m\) and \(\lambda\) accordingly: \(d \sin \theta = 2 (600 \times 10^{-9})\)

Apply the bright fringe formula for the unknown wavelength

For the unknown wavelength, we have the third-order bright fringe, so we set the value of \(m\) accordingly, while using a variable, say \(\lambda_u\), for the unknown wavelength: \(d \sin \theta = 3 \lambda_u\)

Equate both equations since the fringe appears at the same position

Since both fringes appear at the same position, we can equate both equations: \(2 (600 \times 10^{-9}) = 3 \lambda_u\)

Solve for the unknown wavelength

Now all we have to do is solve for \(\lambda_u\): \(\lambda_u = \frac{2 (600 \times 10^{-9})}{3}\) \(\lambda_u = 400 \times 10^{-9}\) Converting back to nanometers, we have: \(\lambda_u = 400\) nm So the value of the unknown wavelength is \(400\) nm.

Key concepts

Wavelength Calculation

Understanding how to calculate the wavelength of light is a fundamental aspect of optics. Wavelength is the distance between two consecutive peaks or troughs in a wave. It is usually denoted by the Greek letter \( \lambda \) and is measured in meters, or more commonly for light, in nanometers (nm).

To calculate the wavelength in a diffraction grating scenario, one must use the formula \( d \sin \theta = m \lambda \), where \( d \) is the grating spacing, \( \theta \) is the angle of the diffraction, and \( m \) is the order of the bright fringe. A bright fringe is a point of constructive interference, where the light waves from different slits arrive in phase, reinforcing one another to create a bright line. By rearranging the equation, \( \lambda = \frac{d \sin \theta}{m} \), the calculation of the wavelength becomes feasible if \( d \) and \( \theta \) are known and the order \( m \) of the bright fringe is observed.

In the given exercise, the wavelength for the second-order bright fringe is compared to an unknown wavelength in the third-order bright fringe, allowing one to solve for the unknown by equating the two equations derived from the same \( \theta \) and \( d \) parameters. Thus, wavelength calculation is a key tool for understanding diffraction patterns in physics.

Interference Patterns

Interference patterns are beautiful and intricate results of waves overlapping and combining with each other. When light waves pass through a diffraction grating, which consists of many equally spaced slits, they spread out and overlap to produce a pattern of bright and dark regions. Constructive interference, where wave peaks align, results in the bright fringes, while destructive interference, where peaks align with troughs, causes dark regions.

The positions of these bright and dark fringes depend on the wavelength of the light and the geometry of the grating. Interference patterns are not random; they follow precise mathematical rules, allowing for the precise measurement of wavelengths. These patterns form the basis for various applications, including spectroscopy, which is used to analyze the composition of light from stars and other sources.

The students are encouraged to visualize these patterns and think about what happens to light at the microscopic level. Understanding the formation of these patterns leads to a deeper comprehension of wave behavior and the characteristics of light.

Physics Problem-Solving

Solving physics problems often requires a systematic approach, such as identifying knowns and unknowns, selecting the appropriate equations, and algebraically manipulating those equations to find a solution. In the context of the given diffraction grating problem, several steps lead to the answer.

Firstly, the problem is approached by understanding the relationship between the variables involved, in this case, using the bright fringe formula. Next, comes the application of the same formula for both known and unknown wavelengths where equations are set up according to the order of bright fringes. Equating these and solving for the unknown variable follows, which is a common technique in physics problem-solving called simultaneous equations.

By breaking down the problem into an orderly sequence of steps, complex problems become manageable. It's advisable to encourage students to adopt a meticulous and patient approach, often reassessing each step for accuracy. This methodical process not only leads to the correct answer but also strengthens understanding and builds confidence in problem-solving skills.