Question

Plane light waves are incident on a single slit of width \(2.00 \mathrm{~cm} .\) The second dark fringe is observed at \(43.0^{\circ}\) from the central axis. What is the wavelength of the light?

Short answer

Answer: The wavelength of the light is approximately 14,500 nm.

Step-by-step solution

Identify relevant information

We are given the following information: - Width of the slit (a) = 2.00 cm - Angle to the second dark fringe (\(\theta\)) = \(43.0^{\circ}\) First, we need to convert the slit width to its SI unit (meters): a (in meters) = 2.00 × 10⁻² m #step_2#

Identify the formula for single-slit diffraction

To find the wavelength, we will use the formula for the angle to a dark fringe in a single-slit diffraction pattern: a * sin(θ) = m * λ where a is the slit width, θ is the angle to the dark fringe, m is the order of the dark fringe, and λ is the wavelength of light. In this case, we are looking for the second dark fringe, so m = 2. #step_3#

Solve for the wavelength of light

Rearrange the formula to solve for λ: λ = (a * sin(θ)) / m Now plug in the values for a, θ, and m: λ = (2 × 10⁻² m * sin(43.0°)) / 2 Computing the result: λ ≈ 1.45 × 10⁻² m #step_4#

Convert the result back to convenient units

The answer is given in meters, but it is more common to express wavelengths in nanometers (nm). To convert the result to nanometers, multiply by 1 billion, or 10⁹: λ ≈ 1.45 × 10⁻² m × 10⁹ nm/m ≈ 14,500 nm Therefore, the wavelength of the light is approximately 14,500 nm.

Key concepts

Diffraction Pattern

When light waves encounter an obstacle or pass through a narrow slit, they bend around it in a phenomenon known as diffraction. The result is a pattern on a screen or any observational medium that consists of regions of constructive and destructive interference, producing bright and dark fringes. The central maximum is the brightest and largest fringe located at the center of the pattern, and it is flanked by alternating dark and bright fringes. Understanding diffractive patterns is crucial because it allows us to determine the properties of light, including its wavelength.

The intricate structure of the diffraction pattern depends on the wavelength of the light and the size of the slit. Larger wavelengths result in more pronounced diffraction, creating wider patterns. The variation in intensity within these patterns is a direct consequence of the wave nature of light, where waves interfere with each other. The dark and bright regions correspond to destructive and constructive interference, respectively. By analyzing these patterns, physicists can delve into the behavior of light as it interacts with small apertures, allowing us not only to explore fundamental principles of wave optics but also to apply this knowledge in creating optical instruments and technologies.

Wavelength Calculation

Calculating the wavelength of light is important for understanding its behavior and interactions with matter. The equation derived from the principle of single-slit diffraction: \[\begin{equation}a \cdot \sin(\theta) = m \cdot \lambda\end{equation}\]is a fundamental tool for this purpose. Here,

• \(a\) represents the width of the slit,
• \(\theta\) is the angle at which a dark fringe occurs,
• \(m\) is the order number of the fringe (an integer value), and
• \(\lambda\) is the wavelength of light being used.

For accurate wavelength calculations, it's important to convert all the measurements into standard units, usually meters for the width and radians or degrees for the angle. The equation can be rearranged to solve for the wavelength (\(\lambda\)), allowing for a practical determination of this fundamental characteristic of light by measuring the angle and knowing the slit width and the order of the dark fringe.

Physics of Light Waves

The physics of light waves encompasses a broad range of concepts, but at its heart is the understanding that light behaves both as a particle and a wave. This dual nature is evident in diffraction and interference patterns. Light waves are transverse waves, which means they oscillate perpendicular to the direction of propagation, and have characteristics of wavelength, frequency, and amplitude.

Wave-Particle Duality

While the study of diffraction patterns underscores the wave aspect of light, it's essential to recognize that in other phenomena light displays particle-like properties, such as in the photoelectric effect. This wave-particle duality is central to quantum mechanics and has profound implications for how we interpret the nature of light and other quantum particles.

Speed, Wavelength, and Frequency

In a vacuum, all light waves travel at the same speed—the speed of light, denoted by \(c\). However, different types of light have different wavelengths and frequencies. The relationship between speed (\(c\)), wavelength (\(\lambda\)), and frequency (\(f\)) is given by \(c = \lambda \times f\). Understanding this relationship is crucial for various applications, including telecommunications, medicine, and astrophysics, where properties of light waves are exploited to obtain information about the world around us and beyond.