Problem 8 - Applied Physics

Question

$$ \begin{aligned} &c=3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} \\ &\lambda=9.23 \mathrm{~km} \\ &f=? \end{aligned} $$

Step-by-Step Solution

Answer

The frequency \( f \) is approximately \( 3.25 \times 10^{4} \, \text{Hz} \).

Step 1: Identify the Formula

The formula to find frequency \( f \) when you have the speed of light \( c \) and wavelength \( \lambda \) is: \[ f = \frac{c}{\lambda} \] where \( f \) is the frequency, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

Step 2: Convert Units

The wavelength \( \lambda \) is given in kilometers, so it needs to be converted to meters to use the formula. \( 1 \, \text{km} = 1000 \, \text{m} \), so \( \lambda = 9.23 \, \text{km} = 9.23 \times 10^{3} \, \text{m} \).

Step 3: Substitute Values into Formula

Insert the given values for \( c \) and the converted \( \lambda \) into the formula. \[ f = \frac{3.00 \times 10^{8} \, \text{m/s}}{9.23 \times 10^{3} \, \text{m}} \]

Step 4: Calculate

Perform the division to find the frequency. \[ f = \frac{3.00 \times 10^{8}}{9.23 \times 10^{3}} \approx 3.25 \times 10^{4} \, \text{Hz} \]

Step 5: Conclusion

Thus, the frequency is approximately \( 3.25 \times 10^{4} \, \text{Hz} \) or \( 32,500 \, \text{Hz} \).

Key Concepts

Speed of Light

Understanding the speed of light is crucial for solving physics problems related to electromagnetic waves. The speed of light, denoted by the symbol \( c \), is a fundamental constant in physics with an approximate value of \( 3.00 \times 10^8 \mathrm{~m/s} \). This constant is the fastest speed at which information or matter can travel in the universe.

The speed of light is especially important when dealing with electromagnetic waves, as it links the wave's frequency and its wavelength. This relationship is expressed by the equation \( c = \lambda \times f \), where \( \lambda \) is wavelength and \( f \) is frequency. Understanding these relationships allows for practical applications, like calculating how radio waves travel or how light behaves.

Remember:

• The speed of light is constant for all electromagnetic waves in a vacuum.
• Light travels slightly slower through other materials like water or glass.
• It's a critical component in equations involving wave frequency and wavelength.

Wavelength Conversion

In problem-solving, wavelength conversion is often necessary to ensure that all units are compatible. Wavelengths are frequently given in units like kilometers, which need to be converted into meters to use standard scientific formulas.

Since 1 kilometer equals 1000 meters, the conversion is straightforward: multiply the value in kilometers by 1000 to get the value in meters. For example, in the original exercise, a wavelength of 9.23 km is converted to 9.23 x 10³ meters.

To convert:

• Identify the original unit of measurement.
• Use the conversion factor (e.g., 1 km = 1000 m).
• Multiply to change to meters if necessary.

Converting to meters is important because the speed of light is measured in meters per second (m/s), and using consistent units throughout calculations helps to avoid errors.

Always ensure that all units match when performing any kind of calculation involving speed, distance, or frequency.

Physics Problem Solving

Problem-solving in physics involves understanding the core concepts and applying appropriate formulas. In the context of electromagnetic waves, it is necessary to identify the relationship between frequency, wavelength, and the speed of light. This often entails converting units and substituting numerical values into the formula.

The steps that are commonly followed include:

• Identify what is given in the problem (e.g., speed, wavelength) and what needs to be found (e.g., frequency).
• Convert any necessary units to match those used in the formulas.
• Substitute the known values into the relevant equation.
• Solve the equation, ensuring attention to detail to calculate accurately.

By following these steps, students can systematically approach physics problems. Detailed understanding of each concept involved in a problem can greatly simplify the solution process and enhance comprehension.

Electromagnetic Waves

Electromagnetic waves are a central topic in physics, encompassing phenomena like radio waves, microwaves, visible light, and X-rays. They are waves of energy that can travel through a vacuum or other medium and are characterized by their wavelength and frequency.

These waves have unique properties:

• They travel at the speed of light in a vacuum.
• They do not require a medium to propagate, unlike sound waves.
• Their wavelength and frequency are inversely proportional, which means as one increases, the other decreases.

Understanding electromagnetic waves is fundamental for many modern technologies, from communication devices to medical imaging.

The study of these waves helps explain how different frequencies are used for various technologies. For example, radio frequencies are lower in the electromagnetic spectrum, allowing them to travel long distances. By contrast, higher frequencies like those of X-rays have much shorter wavelengths and more energy.