Problem 5 - Applied Physics

Question

$$ \text { Find the frequency of electromagnetic radiation with energy } 5.50 \times 10^{-26} \mathrm{~J} \text { . } $$

Step-by-Step Solution

Answer

The frequency of the electromagnetic radiation is \(8.30 \times 10^7 \text{ Hz}\).

Step 1: Recall the Energy-Frequency Formula

To find the frequency of electromagnetic radiation from its energy, we use the formula \(E = h \cdot f\), where \(E\) is the energy, \(h\) is the Planck's constant \(6.626 \times 10^{-34} \text{ J s}\), and \(f\) is the frequency. This formula relates the energy of photons to their frequency.

Step 2: Rearrange the Formula to Isolate Frequency

Rearrange the equation \(E = h \cdot f\) to solve for the frequency \(f\). This gives \(f = \frac{E}{h}\). This rearrangement allows us to plug in the energy value and Planck's constant to find the frequency.

Step 3: Substitute Values into the Equation

Substitute the given energy \(E = 5.50 \times 10^{-26} \text{ J}\) and Planck's constant \(h = 6.626 \times 10^{-34} \text{ J s}\) into the equation \(f = \frac{E}{h}\). This gives \(f = \frac{5.50 \times 10^{-26}}{6.626 \times 10^{-34}}\).

Step 4: Calculate the Frequency

Perform the calculation \(f = \frac{5.50 \times 10^{-26}}{6.626 \times 10^{-34}}\). Divide \(5.50\) by \(6.626\), and subtract the exponents \(-26\) and \(-34\) to determine \(f = 8.30 \times 10^7 \text{ Hz}\). This result gives us the frequency in hertz.

Key Concepts

Energy-Frequency Relationship

Electromagnetic radiation, like light, carries energy. This energy is connected to the frequency of the waves. How? Through the formula: \(E = h \cdot f\). Here, \(E\) stands for energy, \(h\) is Planck's constant, and \(f\) is the frequency of the electromagnetic waves.
This relationship tells us that when frequency increases, energy also increases. Conversely, if you know the energy of electromagnetic radiation, you can determine its frequency by rearranging the formula to \(f = \frac{E}{h}\).
Understanding this relationship is crucial:

• The larger the frequency, the more energetic the radiation.
• High-frequency radiation (like X-rays) carries more energy than low-frequency radiation (like radio waves).

This energy-frequency relationship is a fundamental concept in physics and helps explain how electromagnetic waves interact with matter.

Planck's Constant

Planck's constant, denoted as \(h\), is a fundamental constant in quantum mechanics. Its value is approximately \(6.626 \times 10^{-34} \text{ J s}\). It serves as a bridge between the wave and particle natures of light.
Plank's constant:

• Is essential in calculating photon energy.
• Works as a proportionality factor in the energy-frequency relationship of electromagnetic radiation.

To glimpse further into Planck's constant, think about its role:

• It's tiny because energy at atomic scales is typically small.
• Controls the smallest packet of energy that can be emitted or absorbed as electromagnetic radiation - the photon.

Understanding Planck's constant helps unravel the quantum nature of electromagnetic radiation.

Photon Energy

Photons are packets of electromagnetic energy. The energy of a photon is directly related to its frequency, as described by the equation \(E = h \cdot f\).
This simple relationship reveals:

• Photon energy increases with increasing frequency.
• The energy of each photon is a tiny but significant amount, crucial for quantum physics.

For example:

• A higher-energy gamma-ray photon has a higher frequency than an infrared photon.
• Every color of light corresponds to photons with specific energy based on their frequency.

Photon energy is essential in understanding phenomena in fields like spectroscopy, where it is used to analyze how light interacts with matter.