Calculating The Wavelength Of Yellow Light An Example Problem
Light, an essential aspect of our existence, exhibits a dual nature, behaving as both a wave and a particle. This wave-particle duality is a fundamental concept in physics, and understanding the wave nature of light allows us to explore its properties, such as wavelength and frequency. In this article, we will delve into the concept of wavelength, specifically focusing on yellow light, and calculate its wavelength given its frequency. This exploration will not only provide a numerical answer but also deepen our understanding of the electromagnetic spectrum and the behavior of light.
Light, in its wave form, is characterized by several key parameters, including wavelength, frequency, and speed. Wavelength, denoted by the Greek letter lambda (λ), represents the distance between two consecutive crests or troughs of a wave. It is typically measured in meters (m) or nanometers (nm). Frequency, denoted by the letter f, represents the number of wave cycles that pass a given point per unit time, usually measured in Hertz (Hz), where 1 Hz equals one cycle per second. The speed of light, denoted by the letter c, is a fundamental constant in physics, approximately equal to 3.0 x 10^8 meters per second (m/s) in a vacuum. These three parameters are interconnected by a fundamental equation that governs the behavior of all electromagnetic waves, including light.
The relationship between wavelength, frequency, and the speed of light is described by the following equation:
c = λf
Where:
- c is the speed of light (approximately 3.0 x 10^8 m/s)
- λ is the wavelength (in meters)
- f is the frequency (in Hertz)
This equation is a cornerstone in understanding the behavior of light and other electromagnetic waves. It reveals an inverse relationship between wavelength and frequency: as frequency increases, wavelength decreases, and vice versa. The speed of light, c, remains constant, acting as the proportionality constant between wavelength and frequency. This relationship is crucial for calculating the wavelength of light if its frequency is known, or vice versa. In our case, we are given the frequency of yellow light and asked to determine its wavelength, a task we will undertake using this very equation.
The problem at hand presents a specific scenario: determining the wavelength of yellow light given its frequency. We are provided with the frequency of yellow light, which is $5.2 imes 10^{14} Hz$, and the speed of light, which is a constant value of $3.0 imes 10^8 m/s$. Our goal is to calculate the wavelength (λ) of this yellow light using the relationship between speed, frequency, and wavelength. This calculation is not merely an academic exercise; it demonstrates the practical application of physics principles in understanding the characteristics of light we perceive every day. Yellow light, like all colors of the visible spectrum, has a specific wavelength range, and determining this wavelength helps us to understand its place within the electromagnetic spectrum. Furthermore, this type of calculation is essential in various fields, including optics, telecommunications, and spectroscopy, where the properties of light are critical.
To solve this problem, we will use the equation c = λf, which, as discussed earlier, links the speed of light, wavelength, and frequency. The challenge here is to rearrange this equation to solve for the unknown variable, which in our case is the wavelength (λ). This involves a simple algebraic manipulation to isolate λ on one side of the equation. Once we have rearranged the equation, we can then substitute the given values for the speed of light (c) and the frequency (f) of the yellow light. This will give us a numerical value for the wavelength, which will initially be in meters. However, the problem asks for the answer in proper scientific notation, which is a standard way of expressing very large or very small numbers. We will need to convert our result into scientific notation, ensuring that the answer is presented in the correct format and units. This process will demonstrate our understanding of both the physics concepts and the mathematical techniques required to solve the problem.
To determine the wavelength of the yellow light, we start with the fundamental equation that relates the speed of light (c), wavelength (λ), and frequency (f):
c = λf
Our objective is to find the wavelength (λ), so we need to rearrange the equation to solve for λ. We can do this by dividing both sides of the equation by the frequency (f):
λ = c / f
Now we have the equation in the form we need, with the wavelength (λ) isolated on one side. The next step is to substitute the given values into the equation. We are given that the frequency (f) of the yellow light is $5.2 imes 10^{14} Hz$ and the speed of light (c) is $3.0 imes 10^8 m/s$. Substituting these values into the equation, we get:
λ = ($3.0 imes 10^8 m/s$) / ($5.2 imes 10^{14} Hz$)
Now, we perform the division:
λ ≈ 5.769 × 10^-7 m
This result gives us the wavelength in meters. However, the problem requires the answer in proper scientific notation. Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. Our current result, 5.769 × 10^-7 m, is already in scientific notation, so no further conversion is needed in this case. The wavelength of the yellow light is approximately $5.769 imes 10^{-7} meters$. This value falls within the typical wavelength range for yellow light, which is between 570 and 590 nanometers. Therefore, our calculated value is consistent with the known properties of yellow light.
The final step is to express our calculated wavelength in the format requested by the problem, which is proper scientific notation. As we found in the previous section, the wavelength of the yellow light is approximately $5.769 imes 10^{-7} meters$. This value is already in scientific notation, which is a convenient way to represent very small or very large numbers. Scientific notation consists of a number between 1 and 10 (the coefficient) multiplied by a power of 10 (the exponent). In our case, the coefficient is 5.769, and the exponent is -7. This notation allows us to avoid writing out many zeros and makes it easier to compare numbers of different magnitudes.
The problem statement asks for the answer in the form [?] × 10[?] m. Our calculated value fits this format perfectly. The coefficient is 5.769, and the exponent is -7. Therefore, the final answer, expressed in proper scientific notation, is:
This result indicates that the wavelength of the yellow light is a very small fraction of a meter, which is characteristic of light waves. The negative exponent signifies that we are dealing with a number less than 1. This answer is not just a numerical result; it provides insight into the scale of light waves and their place in the electromagnetic spectrum. It also demonstrates the importance of using appropriate units and notation when expressing physical quantities. Scientific notation is a crucial tool in physics and other scientific disciplines, allowing for clear and concise communication of numerical values, regardless of their size.
In this exploration, we successfully calculated the wavelength of yellow light given its frequency. We began by understanding the fundamental relationship between the speed of light, wavelength, and frequency, expressed by the equation c = λf. We then applied this knowledge to solve a specific problem, determining the wavelength of yellow light with a frequency of $5.2 imes 10^{14} Hz$. Our calculations led us to a wavelength of approximately $5.769 imes 10^{-7} meters$, which we expressed in proper scientific notation as $5.769 imes 10^{-7} m$. This result aligns with the known properties of yellow light, which falls within the wavelength range of 570 to 590 nanometers.
This exercise underscores the wave nature of light and the importance of understanding the relationships between its properties. Wavelength and frequency are inversely proportional, meaning that as one increases, the other decreases, a principle that governs the behavior of all electromagnetic waves. The speed of light, a fundamental constant, acts as the bridge connecting these two properties. By manipulating the equation c = λf, we can determine any one of these properties if the other two are known. This ability is not just an academic exercise; it has practical applications in various fields, including optics, telecommunications, and medical imaging. The understanding of light's properties allows us to develop technologies that rely on its behavior, from the lenses in our glasses to the fiber optic cables that transmit data across the globe.
Furthermore, this exploration highlights the significance of scientific notation in expressing and working with very small or very large numbers. Scientific notation provides a concise and clear way to represent these numbers, making calculations and comparisons easier. In the case of light wavelengths, which are typically on the order of nanometers (billionths of a meter), scientific notation is essential for avoiding cumbersome strings of zeros. The ability to express physical quantities in appropriate units and notation is a critical skill in physics and other scientific disciplines. As we continue to explore the world around us, understanding the properties of light and the tools we use to describe them will remain fundamental to our scientific endeavors.
