Photon Energy Calculation: Frequency & Planck's Constant

Nov 7, 2025 by Admin 57 views

Hey everyone! Let's dive into the fascinating world of quantum physics and tackle a problem involving photons, frequency, and energy. We're going to calculate the energy of a photon, and it's actually pretty straightforward once you understand the key concepts and the formula involved.

Understanding the Key Concepts

Before we jump into the calculation, let's quickly review the important concepts:

Photons: Think of photons as tiny packets of light or electromagnetic radiation. They are fundamental particles, meaning they aren't made up of anything smaller.
Energy (E): Energy, in simple terms, is the ability to do work. In physics, we often talk about energy in units of Joules (J).
Frequency (ν): Frequency refers to how many waves pass a certain point in a given amount of time, usually measured in Hertz (Hz). One Hertz means one cycle per second.
Planck's Constant (h): This is a fundamental constant in quantum mechanics, denoted by the symbol 'h'. It links the energy of a photon to its frequency. The value of Planck's constant is approximately 6.63 x 10^-34 Joule-seconds (J⋅s).

The Formula: Connecting Energy and Frequency

The magic formula that connects these concepts is:

E = hν

Where:

E is the energy of the photon (in Joules)
h is Planck's constant (approximately 6.63 x 10^-34 J⋅s)
ν is the frequency of the photon (in Hertz)

This equation, guys, is a cornerstone of quantum mechanics. It tells us that the energy of a photon is directly proportional to its frequency. The higher the frequency, the higher the energy, and vice-versa. Now that we've got our concepts down, let's apply this to the specific problem.

Problem Setup

Okay, let's get to the problem at hand. We're given the following information:

Frequency (ν) = 3.6 x 10^15 Hz
Planck's constant (h) = 6.63 x 10^-34 J⋅s

And we need to find the energy (E) of the photon. See? It's like a puzzle where we have most of the pieces already!

Applying the Formula: Step-by-Step

Now it’s time to plug the given values into our formula, E = hν.

Write down the formula: E = hν
Substitute the values: E = (6.63 x 10^-34 J⋅s) x (3.6 x 10^15 Hz)
Multiply the numbers: E = 23.868 x 10^(-34+15) J
Simplify the exponent: E = 23.868 x 10^-19 J
Express in scientific notation: E = 2.3868 x 10^(-19+1) J
Round the answer: E ≈ 2.4 x 10^-18 J

So, there you have it! The energy of the photon is approximately 2.4 x 10^-18 Joules. Remember, it's crucial to keep track of your units during the calculation. Planck's constant has units of Joule-seconds (J⋅s), and frequency is in Hertz (Hz), which is equivalent to 1/seconds (s^-1). When you multiply them, the seconds cancel out, leaving you with Joules (J), which is the correct unit for energy.

Analyzing the Answer Choices

Now, let's look at the answer choices provided and see which one matches our calculated answer:

A. 1.8 x 10^-49 J B. 2.4 x 10^-19 J C. 1.8 x 10^-18 J D. 2.4 x 10^-18 J

We can see that option D, 2.4 x 10^-18 J, matches our calculated energy. Therefore, that's our correct answer!

Why Other Options Are Incorrect

It's always a good idea to think about why the other options are incorrect. This helps solidify your understanding of the concepts.

Option A (1.8 x 10^-49 J): This value is extremely small. The exponent is -49, which means this number is much, much smaller than 10^-18 J. This suggests there might have been an error in calculation, perhaps involving incorrect exponents or a misunderstanding of the formula.
Option B (2.4 x 10^-19 J): This value is close to our answer in terms of the digits (2.4), but the exponent is different (-19 instead of -18). This indicates a possible mistake in handling the exponents during the multiplication or simplification steps.
Option C (1.8 x 10^-18 J): This value has the correct exponent (-18) but a different leading number (1.8 instead of 2.4). This could result from an arithmetic error when multiplying Planck's constant and the frequency. Always double-check your calculations, guys!

Common Mistakes to Avoid

Calculating photon energy is a fundamental physics problem, and like any calculation, there are some common pitfalls to watch out for:

Exponent Errors: As we saw in the analysis of incorrect options, mistakes with exponents are super common. When multiplying numbers in scientific notation, you multiply the coefficients and add the exponents. Make sure you're doing this correctly!
Unit Confusion: Always keep track of your units. Using the wrong units can lead to completely incorrect answers. Remember, energy is measured in Joules (J), frequency in Hertz (Hz), and Planck's constant in Joule-seconds (J⋅s).
Calculator Errors: Calculators are helpful, but they can also be a source of errors if you don't use them carefully. Double-check your inputs, especially when dealing with scientific notation and exponents.
Forgetting the Formula: It seems obvious, but forgetting the formula E = hν is a surefire way to get the problem wrong. Make sure you memorize it and understand what each term represents.
Rounding Errors: Rounding too early in the calculation can affect your final answer. It's generally best to keep as many digits as possible during the calculation and only round at the very end.

Real-World Applications

Okay, so we've calculated the energy of a photon, but why is this important? Well, understanding photon energy has tons of applications in the real world. Let's look at a few:

Photosynthesis: Plants use photons from sunlight to perform photosynthesis, the process of converting light energy into chemical energy. The energy of the photons determines which wavelengths of light plants can use.
Medical Imaging: X-rays and gamma rays are high-energy photons used in medical imaging techniques like X-rays and PET scans. The energy of these photons allows them to penetrate tissues and create images of bones and organs.
Solar Panels: Solar panels convert photons from sunlight into electricity. The energy of the photons determines the amount of electricity that can be generated.
Laser Technology: Lasers emit photons with very specific energies and wavelengths. This property is used in a wide range of applications, from laser pointers to laser surgery.
Spectroscopy: Scientists use spectroscopy to analyze the light emitted or absorbed by substances. By studying the energies of the photons, they can determine the composition and properties of the substance.

These are just a few examples, guys. Photon energy plays a critical role in many areas of science and technology. So, understanding this concept is not just about solving physics problems; it's about understanding the world around us.

Practice Problems

Alright, to really master this, let's try a few practice problems. Remember the formula E = hν and watch out for those exponents!

What is the energy of a photon with a frequency of 5.0 x 10^14 Hz?
A photon has an energy of 3.3 x 10^-19 J. What is its frequency?
If the wavelength of a photon is 450 nm, what is its energy? (Hint: You'll need to use the relationship between frequency, wavelength, and the speed of light.)

Work through these, and you'll be a photon energy pro in no time! Try working them out on your own first, and then you can always search online for solutions to check your work.

Conclusion

Calculating the energy of a photon using its frequency and Planck's constant is a fundamental skill in physics. By understanding the concepts, the formula E = hν, and avoiding common mistakes, you can confidently tackle these types of problems. Plus, knowing about photon energy opens the door to understanding so many cool real-world applications. Keep practicing, keep exploring, and keep learning, everyone! Physics is awesome, and you've got this!