Energy/Frequency/Wavelength Worksheet Answer Key Explained
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The relationship between energy, frequency, and wavelength is fundamental to our understanding of the electromagnetic spectrum. These concepts are often explored in high school and introductory college-level physics courses, and worksheets that cover these topics help students solidify their knowledge through practice. In this article, we will explain the answers to a typical Energy/Frequency/Wavelength worksheet, enhancing your understanding of these interrelated topics.
Understanding the Basics
Before diving into the worksheet answers, it's essential to understand the key formulas that govern the relationship between energy (E), frequency (ν), and wavelength (λ).
Key Formulas
-
Energy and Frequency
- The energy of a photon is directly proportional to its frequency: [ E = hν ] Where:
- (E) = energy (in joules)
- (h) = Planck's constant ((6.626 \times 10^{-34} , \text{J s}))
- (ν) = frequency (in hertz)
-
Frequency and Wavelength
- Frequency is inversely proportional to wavelength: [ c = νλ ] Where:
- (c) = speed of light ((3.00 \times 10^8 , \text{m/s}))
- (λ) = wavelength (in meters)
Energy, Frequency, and Wavelength Relationship
From the above formulas, we can derive that: [ E = \frac{hc}{λ} ] This equation illustrates that energy is inversely proportional to wavelength; as wavelength increases, energy decreases.
Worksheet Answers Explained
Example Problem 1: Finding Energy from Frequency
Question: What is the energy of a photon with a frequency of (5 \times 10^{14} , \text{Hz})?
Solution: Using the formula (E = hν): [ E = (6.626 \times 10^{-34} , \text{J s})(5 \times 10^{14} , \text{Hz}) = 3.313 \times 10^{-19} , \text{J} ] This calculation shows that a photon with a frequency of (5 \times 10^{14} , \text{Hz}) has an energy of approximately (3.31 \times 10^{-19} , \text{J}).
Example Problem 2: Finding Wavelength from Frequency
Question: What is the wavelength of a photon with a frequency of (1 \times 10^{15} , \text{Hz})?
Solution: Using the formula (c = νλ): [ λ = \frac{c}{ν} = \frac{3.00 \times 10^8 , \text{m/s}}{1 \times 10^{15} , \text{Hz}} = 3.00 \times 10^{-7} , \text{m} ] Thus, the wavelength of a photon with a frequency of (1 \times 10^{15} , \text{Hz}) is approximately (3.00 \times 10^{-7} , \text{m}), or (300 , \text{nm}), which falls in the ultraviolet range of the electromagnetic spectrum.
Example Problem 3: Finding Frequency from Wavelength
Question: Determine the frequency of a photon with a wavelength of (500 , \text{nm}).
Solution: First, convert wavelength to meters: [ 500 , \text{nm} = 500 \times 10^{-9} , \text{m} ] Now, use the formula (ν = \frac{c}{λ}): [ ν = \frac{3.00 \times 10^8 , \text{m/s}}{500 \times 10^{-9} , \text{m}} = 6.00 \times 10^{14} , \text{Hz} ] The frequency of a photon with a wavelength of (500 , \text{nm}) is therefore (6.00 \times 10^{14} , \text{Hz}).
Example Problem 4: Energy from Wavelength
Question: Calculate the energy of a photon with a wavelength of (650 , \text{nm}).
Solution: Convert wavelength to meters: [ 650 , \text{nm} = 650 \times 10^{-9} , \text{m} ] Now, use the energy formula: [ E = \frac{hc}{λ} = \frac{(6.626 \times 10^{-34} , \text{J s})(3.00 \times 10^8 , \text{m/s})}{650 \times 10^{-9} , \text{m}} \approx 3.08 \times 10^{-19} , \text{J} ] So, the energy of a photon with a wavelength of (650 , \text{nm}) is approximately (3.08 \times 10^{-19} , \text{J}).
Summary Table of Key Relationships
Below is a summary table that recaps our discussions and can serve as a handy reference.
<table> <tr> <th>Quantity</th> <th>Formula</th> <th>Unit</th> </tr> <tr> <td>Energy</td> <td>E = hν</td> <td>Joules (J)</td> </tr> <tr> <td>Frequency</td> <td>ν = c / λ</td> <td>Hertz (Hz)</td> </tr> <tr> <td>Wavelength</td> <td>λ = c / ν</td> <td>meters (m)</td> </tr> <tr> <td>Energy from Wavelength</td> <td>E = hc / λ</td> <td>Joules (J)</td> </tr> </table>
Important Notes
"Always remember that energy is inversely related to wavelength. As wavelength decreases (moving from radio waves towards gamma rays), energy increases significantly."
By understanding these core principles and practicing with worksheets, students can develop a comprehensive understanding of how energy, frequency, and wavelength interconnect in the realm of electromagnetic radiation. This knowledge lays the groundwork for deeper explorations into areas such as quantum mechanics and wave-particle duality.