Problem 3 - Applied Physics

Question

$$ \begin{aligned} &c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} \\ &f=9.70 \times 10^{11} \mathrm{~Hz} \\ &\lambda=? \end{aligned} $$

Step-by-Step Solution

Answer

The wavelength is approximately \(3.09 \times 10^{-4}\) meters.

Step 1: Identify the Formula for Wavelength

The relationship between the speed of light \( c \), frequency \( f \), and wavelength \( \lambda \) is given by the formula: \[ \lambda = \frac{c}{f} \]

Step 2: Substitute the Known Values

We have been given the speed of light \( c = 3.00 \times 10^8 \) m/s and the frequency \( f = 9.70 \times 10^{11} \) Hz. Substitute these values into the formula: \[ \lambda = \frac{3.00 \times 10^8}{9.70 \times 10^{11}} \]

Step 3: Calculate the Wavelength

Perform the division to find the wavelength: \[ \lambda = \frac{3.00 \times 10^8}{9.70 \times 10^{11}} \approx 3.09 \times 10^{-4} \text{ meters} \]

Step 4: Interpret the Result

The wavelength \( \lambda \) of the wave is approximately \( 3.09 \times 10^{-4} \) meters, or 309 micrometers.

Key Concepts

Speed of Light

The speed of light is fundamental in the universe. It is the speed at which all massless particles, such as photons, travel in a vacuum. The generally accepted value of the speed of light is \( c = 3.00 \times 10^{8} \, \text{m/s} \).
The speed of light is integral to understanding many aspects of physics, including the theory of relativity and electromagnetic theory. Because light travels so fast, it allows us to perceive events happening in real time from various distances. This speed is constant and serves as a critical connector between various physical concepts like wavelength and frequency. In problems related to wave equations, knowing the speed of light lets us find either frequency or wavelength as long as the other component is known.

Frequency and Wavelength Relationship

The connection between frequency and wavelength is crucial in physics wave studies. Frequency \((f)\) corresponds to the number of oscillations of a wave per second and is measured in hertz \((Hz)\). Wavelength \((\lambda)\), on the other hand, is the physical length of one cycle of the wave, typically measured in meters.
These two properties are inversely related; as one increases, the other decreases. The mathematical expression linking them is given by the formula:
\[ \lambda = \frac{c}{f} \]
This equation shows that wavelength equals the speed of light \((c)\) divided by the frequency \((f)\).

• If frequency increases, wavelength decreases.
• If frequency decreases, wavelength increases.

This relationship explains how energy is propagated at different scales. For instance, high-frequency waves like gamma rays have tiny wavelengths, while low-frequency waves like radio waves have very long wavelengths.

Physics Problem Solving

Approaching a physics problem, like calculating wavelength from frequency and speed, involves logical steps that clarify the process. Here's a concise way to understand problem-solving in physics:

• Identify the formula that relates the quantities in the problem. For our example, we used \( \lambda = \frac{c}{f} \).
• Substitute known values into the formula. This means replacing letters with numbers, ensuring each substitution is consistent with its unit.
• Perform calculations carefully, using a calculator if necessary, to ensure precision.
• Interpret the results to give them a physical meaning, understanding what the values actually represent in real-world terms.

Employing this systematic approach helps in tackling various physics problems efficiently. It's not just about finding the answer but about understanding how each component interacts to lead to the obtained solution.