Calculating Wavelength Of Light In Glass
In the fascinating realm of physics, light exhibits wave-like properties, and understanding its behavior as it travels through different media is crucial. This article delves into the concept of the wavelength of light as it propagates through glass. We will explore the relationship between the speed of light, its frequency, and its wavelength, ultimately determining the wavelength of light with a given frequency traveling through glass. Light, an electromagnetic wave, slows down when it enters a medium like glass. This reduction in speed affects the light's wavelength, while its frequency remains constant. This interplay between speed, frequency, and wavelength is governed by a fundamental equation, which we will explore in detail.
The speed of light in a vacuum is a universal constant, approximately 2.998 x 10^8 m/s. However, when light enters a medium like glass, it interacts with the atoms in the glass, causing it to slow down. The extent to which light slows down is determined by the refractive index of the medium. Glass, being a denser medium than air or vacuum, has a refractive index greater than 1, which means light travels slower in glass. The frequency of light, on the other hand, is a property of the light source itself and does not change as light moves from one medium to another. Frequency represents the number of wave cycles that pass a point in a given amount of time, typically measured in Hertz (Hz), which is cycles per second.
Understanding the wavelength of light in different media is essential for various applications, including optics, telecommunications, and material science. For instance, the design of lenses and optical fibers relies heavily on the principles of refraction and the changes in wavelength that occur as light passes through different materials. In telecommunications, the wavelength of light used in fiber optic cables is carefully chosen to minimize signal loss and maximize transmission efficiency. Furthermore, the interaction of light with materials at different wavelengths provides valuable information about the material's structure and properties, which is crucial in material science research.
To calculate the wavelength of light (λ) in a medium, we utilize the fundamental relationship between speed (v), frequency (f), and wavelength: v = fλ. This equation is a cornerstone of wave physics, applicable not only to light waves but also to other types of waves, such as sound waves. It highlights the direct proportionality between speed and wavelength when the frequency is constant, and the inverse proportionality between frequency and wavelength when the speed is constant. To find the wavelength, we simply rearrange the formula to: λ = v / f. This rearranged formula allows us to calculate the wavelength if we know the speed of light in the medium and its frequency.
In this specific problem, we are given that the speed of light in glass (v) is 2.0 x 10^8 m/s, and the frequency (f) of the light is 3.1 x 10^14 Hz. Plugging these values into the formula, we get: λ = (2.0 x 10^8 m/s) / (3.1 x 10^14 Hz). Performing this calculation will yield the wavelength of the light in meters. It's important to pay close attention to the units involved in the calculation to ensure that the final answer is in the correct units. In this case, meters per second (m/s) divided by Hertz (Hz), which is equivalent to cycles per second, results in meters (m), the standard unit for wavelength.
The calculation itself involves dividing the speed of light in glass by the frequency of the light. The speed of light in glass, 2.0 x 10^8 m/s, represents how fast the light wave is propagating through the glass medium. The frequency, 3.1 x 10^14 Hz, represents how many wave cycles occur per second. By dividing the speed by the frequency, we are essentially finding the length of each wave cycle, which is the wavelength. This calculation demonstrates the practical application of the fundamental relationship between speed, frequency, and wavelength, allowing us to determine the physical dimension of the light wave as it travels through a specific medium.
Let's break down the solution step-by-step to ensure clarity and understanding. First, we identify the given values: The speed of light in glass (v) is 2.0 x 10^8 m/s, and the frequency (f) of the light is 3.1 x 10^14 Hz. These values are crucial inputs for our calculation. It is important to note the units of these values as they are in standard units (meters per second and Hertz), which simplifies the calculation process. If the values were given in different units, we would need to convert them to standard units before proceeding with the calculation. The correct identification and understanding of the given values are essential for accurate problem-solving in physics.
Next, we recall the formula that relates speed, frequency, and wavelength of light: λ = v / f. This formula is the key to solving this problem. It expresses the wavelength as the ratio of the speed of light to its frequency. Understanding the relationship between these quantities is fundamental in wave physics. The formula highlights the inverse relationship between wavelength and frequency; for a constant speed, as the frequency increases, the wavelength decreases, and vice versa. This relationship is crucial for understanding various phenomena, such as the electromagnetic spectrum and the behavior of light in different media.
Now, we substitute the given values into the formula: λ = (2.0 x 10^8 m/s) / (3.1 x 10^14 Hz). This step involves replacing the variables in the formula with their corresponding numerical values. It is important to ensure that the values are substituted correctly and that the units are consistent. Substituting the values correctly is a critical step in obtaining an accurate result. After substitution, we perform the calculation by dividing 2.0 x 10^8 by 3.1 x 10^14. This division will give us the numerical value of the wavelength of light in meters.
Performing the division, we get: λ ≈ 6.45 x 10^-7 m. This is the numerical result of our calculation, representing the wavelength of light in glass under the given conditions. It is a very small value, which is typical for the wavelength of light. This value is in scientific notation, which is a convenient way to express very large or very small numbers. The negative exponent indicates that the wavelength is a fraction of a meter. To better understand this value, we can convert it to nanometers (nm), where 1 nm = 10^-9 m. In nanometers, the wavelength is approximately 645 nm, which falls within the visible light spectrum, specifically in the orange-red region. This result provides valuable insight into the nature of light and its behavior as a wave.
Therefore, the wavelength of the light in glass is approximately 6.45 x 10^-7 m, or 645 nm. However, when we compare this result to the multiple-choice options provided, we need to consider which option is closest to our calculated value. None of the options exactly match our calculated value, indicating a potential need for rounding or an approximation in the given choices. The closest option to our calculated wavelength of light 6.45 x 10^-7 m would be an option reflecting a similar magnitude and scientific notation. Examining the provided choices, we would select the one that is nearest to this value. It's essential to understand that in multiple-choice questions, the correct answer might not always be an exact match but the closest logical option.
Understanding the magnitude of the answer is crucial. A wavelength of light on the order of 10^-7 meters is typical for visible light. This falls within the range of hundreds of nanometers, which aligns with our conversion to 645 nm. This consistency check helps validate our calculation and ensures that the answer is physically reasonable. If we had obtained a drastically different magnitude, such as 10^-23 meters or 10^6 meters, it would indicate a potential error in our calculation or understanding of the problem. Thus, always comparing the result with known physical scales helps in verifying the correctness of the solution.
The process of solving this problem highlights the importance of understanding the fundamental relationships in physics, particularly the relationship between speed, frequency, and wavelength of light. By applying the formula v = fλ, we can determine the wavelength of light in different media, given its speed and frequency. This understanding is crucial for various applications in optics, telecommunications, and material science. Furthermore, the step-by-step solution presented here demonstrates a systematic approach to problem-solving in physics, which involves identifying the given values, recalling the relevant formula, substituting the values, performing the calculation, and interpreting the result. This approach is applicable to a wide range of physics problems and helps in developing a deeper understanding of the subject.
In conclusion, we have successfully determined the wavelength of light traveling through glass using the fundamental relationship between speed, frequency, and wavelength. The calculated wavelength of light is approximately 6.45 x 10^-7 m, which corresponds to 645 nm, falling within the visible light spectrum. This exercise demonstrates the practical application of physics principles and the importance of understanding the wave nature of light. The ability to calculate wavelength of light in different media is essential for various scientific and technological applications, highlighting the significance of this concept in the broader field of physics.
