Calculate Photon Wavelength A Step-by-Step Guide
Hey everyone! Today, we're diving into a fascinating physics problem: calculating the wavelength of a photon given its frequency. This is a classic example of how we can use fundamental physics principles to understand the behavior of light. So, let's get started and break down the problem step by step.
Understanding the Basics: Frequency, Wavelength, and the Speed of Light
Before we jump into the calculations, let's quickly review the key concepts involved. The frequency of a photon (often represented by the symbol f) tells us how many wave cycles pass a given point per second, and it's measured in Hertz (Hz). Think of it like this: if you're watching waves at the beach, the frequency is how many wave crests you see crashing onto the shore every second. A frequency of 4.72 x 10^14 Hz means that 472 trillion wave cycles pass a point each second – pretty fast, huh?
The wavelength (represented by the Greek letter lambda, λ) is the distance between two consecutive crests (or troughs) of a wave. It's essentially the physical length of one complete wave cycle. Wavelength is usually measured in meters (m) or nanometers (nm), where 1 nm is one billionth of a meter (1 x 10^-9 m). Shorter wavelengths correspond to higher-energy light, like ultraviolet or X-rays, while longer wavelengths correspond to lower-energy light, like infrared or radio waves.
Now, here’s the really important part: the frequency and wavelength of light are related by a fundamental constant – the speed of light (represented by the letter c). The speed of light in a vacuum is approximately 2.998 x 10^8 meters per second. This is the ultimate speed limit in the universe, and it connects frequency and wavelength through a simple equation:
c = fλ
This equation tells us that the speed of light is equal to the product of the frequency and the wavelength. It’s like a little formula that unlocks the relationship between these two properties of light. We're going to use this equation to solve our problem, so keep it in mind!
To truly grasp the relationship between these concepts, imagine a string stretched out, and you're wiggling one end to create waves. If you wiggle it faster (higher frequency), the waves will be closer together (shorter wavelength). If you wiggle it slower (lower frequency), the waves will be farther apart (longer wavelength). The speed at which the waves travel down the string (the speed of light in our case) remains constant.
This understanding of frequency, wavelength, and the speed of light forms the bedrock of our ability to calculate the wavelength of a photon. Remember, these aren't just abstract concepts; they're the fundamental properties that govern how light behaves, from the colors we see to the technology we use every day. So, with this knowledge in hand, let's move on to the actual calculation!
Calculating the Wavelength
Okay, guys, now that we've got the basics down, let's tackle the calculation. We're given the frequency of the photon, f = 4.72 x 10^14 Hz, and we know the speed of light, c = 2.998 x 10^8 m/s. Our goal is to find the wavelength, λ. To do this, we'll rearrange our trusty equation c = fλ to solve for λ:
λ = c / f
This rearranged equation tells us that the wavelength is equal to the speed of light divided by the frequency. Simple, right? Now we just need to plug in our values and crunch the numbers.
So, let's substitute the values we know:
λ = (2.998 x 10^8 m/s) / (4.72 x 10^14 Hz)
When you perform this division, you'll get a result in meters. Grab your calculators, folks! The calculation yields:
λ ≈ 6.35 x 10^-7 m
But wait! The answer choices are given in nanometers (nm), not meters. No problem – we just need to convert meters to nanometers. Remember that 1 nm = 1 x 10^-9 m. To convert from meters to nanometers, we multiply by 10^9:
λ ≈ 6.35 x 10^-7 m * (10^9 nm / 1 m)
λ ≈ 635 nm
And there we have it! The wavelength of the photon is approximately 635 nm. This falls within the visible light spectrum, specifically in the orange-red region. So, if we could
