Problem 12 - Applied Physics

Question

$$ \text { Find the frequency of an electromagnetic wave if its wavelength is } 3.25 \times 10^{-8} \mathrm{~m} \text { . } $$

Step-by-Step Solution

Answer

The frequency of the electromagnetic wave is \( 9.23 \times 10^{15} \text{ Hz} \).

Step 1: Recall the Speed of Light Formula

Electromagnetic waves, including light, travel at a constant speed in a vacuum. The speed of light \( c \) is approximately \( 3.00 \times 10^8 \text{ m/s} \). The formula that relates speed, frequency (\( f \)), and wavelength (\( \lambda \)) is \( c = f \times \lambda \).

Step 2: Rearrange the Formula to Solve for Frequency

Since we are looking for the frequency, we need to rearrange the formula to isolate \( f \). This gives us \( f = \frac{c}{\lambda} \).

Step 3: Plug in the Known Values

Now, substitute the known values into the equation. The speed of light \( c \) is \( 3.00 \times 10^8 \text{ m/s} \) and the wavelength \( \lambda \) is given as \( 3.25 \times 10^{-8} \text{ m} \).

Step 4: Carry Out the Division

Calculate the frequency using the rearranged equation: \[ f = \frac{3.00 \times 10^8}{3.25 \times 10^{-8}} \] Divide the numbers: \[ f = 3.00 \div 3.25 \times 10^{8 - (-8)} \] \[ f \approx 0.923 \times 10^{16} \]

Step 5: Adjust the Scientific Notation

Change the result from Step 4 to proper scientific notation: \[ f \approx 9.23 \times 10^{15} \text{ Hz} \] This is the frequency of the electromagnetic wave.

Key Concepts

Frequency Calculation

To determine the frequency of an electromagnetic wave, one of the fundamental relationships we use connects the speed of light, frequency, and wavelength. The formula is:

• \( c = f \times \lambda \)

This equation states that the speed of light \( c \) is equal to the frequency \( f \) multiplied by the wavelength \( \lambda \). Frequency, measured in Hertz (Hz), tells us how many wave cycles pass a point per second. In calculations, rearranging the formula to solve for frequency gives us:

• \( f = \frac{c}{\lambda} \)

This means the frequency can be determined by dividing the speed of light by the wavelength of the wave. This straightforward formula makes frequency calculation a simple division, provided you have the speed of light and the wavelength available.

Wavelength

Wavelength is a key factor in determining the properties of an electromagnetic wave. It is the distance between consecutive peaks of a wave. Wavelength is denoted by the Greek letter \( \lambda \) and is usually measured in meters (m). In the context of electromagnetic waves, wavelength helps determine the wave's energy and color (for visible light). It is inversely proportional to frequency, meaning as wavelength increases, frequency decreases and vice versa. This is because:

• \( f = \frac{c}{\lambda} \)

It's crucial to use the correct units when working with wavelength, often necessitating conversion to meters, especially if dealing with other units like nanometers. Wavelength is especially handy in applications like spectroscopy, where it's used to identify the composition of substances.

Speed of Light

The speed of light, denoted as \( c \), is a constant in a vacuum that measures how fast light travels. Its approximate value is \( 3.00 \times 10^{8} \text{ m/s} \). This constant is essential for many calculations in physics, especially in the study of electromagnetic waves. The idea that light travels at this constant speed across any vacuum was pivotal in the development of Einstein's theory of relativity.

The speed of light allows us to connect frequency and wavelength through its equation:

• \( c = f \times \lambda \)

Given any two of these values, we can calculate the third. Besides physics, knowing the speed of light is useful in astronomy for understanding distances in space, such as expressing light-years.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It's a crucial tool in fields like physics, where extreme values are common. A number in scientific notation is split into two components: a coefficient and a power of ten. For example, \( 3.00 \times 10^{8} \) represents the speed of light. Here, 3.00 is the coefficient, and \( 10^{8} \) indicates how many times the coefficient should be multiplied by ten.

This method simplifies calculations, as the powers of ten can be easily added or subtracted. For division, the exponents are subtracted: when \( c \), \( f \), and \( \lambda \) are expressed in scientific notation, determining frequency through:

• \( f = \frac{3.00 \times 10^{8}}{3.25 \times 10^{-8}} \)

becomes simpler. Keeping numbers in scientific notation throughout prevents errors and maintains precision in large datasets or complex equations.