Question

The laser in an audio CD player uses light with a wavelength of \(7.80 \times 10^{2} \mathrm{~nm} .\) Calculate the frequency of this light.

Short answer

The frequency of the light used in the audio CD player is approximately \(3.85 \times 10^{14}\ \mathrm{Hz}\).

Step-by-step solution

Write down the given information

The wavelength of the laser light is given by: \(\lambda = 7.80 \times 10^2 \ \mathrm{nm}\)

Convert the wavelength to meters

Since the speed of light is in meters per second, we need to convert the given wavelength to meters. This can be done by multiplying the given wavelength by the conversion factor from nanometers to meters: \[1\ \mathrm{nm} = 1\times10^{-9}\ \mathrm{m}\] So, \(\lambda = 7.80 \times 10^2\ \mathrm{nm} \times 1\times10^{-9}\ \frac{\mathrm{m}}{\mathrm{nm}} = 7.80 \times 10^{-7}\ \mathrm{m}\)

Recall the wave equation

The wave equation relates the speed of light (c), the frequency (f), and the wavelength (\(\lambda\)): \[c = f\lambda\] Where, c is the speed of light, approximately \(3\times10^8\ \mathrm{m/s}\), f is the frequency, and \(\lambda\) is the wavelength.

Solve for frequency

We will solve the equation above for the frequency (f) by dividing both sides by the wavelength (\(\lambda\)): \[f = \frac{c}{\lambda}\]

Plug in the values and calculate the frequency

Now, we can plug in the values for the speed of light and the wavelength, and calculate the frequency: \[f = \frac{3 \times 10^8\ \mathrm{m/s}}{7.80 \times 10^{-7}\ \mathrm{m}}\] Calculating the result, we get: \[f \approx 3.85 \times 10^{14}\ \mathrm{Hz}\] So, the frequency of the light used in the audio CD player is approximately \(3.85 \times 10^{14}\ \mathrm{Hz}\).

Key concepts

Wavelength Conversion

Understanding how to convert between different units is an essential skill in physics and everyday life. When dealing with light, the wavelength is often given in nanometers (nm) because light waves are very small. However, most equations, including the wave equation, use meters (m) as the standard unit of length.

Converting from nanometers to meters involves multiplying by a conversion factor. Since there are one billion nanometers in a meter, the conversion factor is \(1\times10^{-9}\) meters per nanometer. Here's how the conversion looks mathematically:

• Given wavelength in nanometers: \(\lambda_{nm}\)
• Wavelength in meters: \(\lambda_m = \lambda_{nm} \times 1\times10^{-9} \, \mathrm{m/nm}\)

Applying this to our laser light example, \(7.80 \times 10^{2} \, \mathrm{nm}\) converts to \(7.80 \times 10^{2} \times 1\times10^{-9}\) meters, or \(7.80 \times 10^{-7}\) meters.
Small conversions like these can make a big difference in calculations and understanding the scale of different phenomena.

Wave Equation

The wave equation is a foundational concept in physics that connects the wavelength (\(\lambda\)), frequency (\(f\)), and the speed of a wave (\(c\)). For electromagnetic waves, which include light, the equation is elegantly simple:

\[c = f\lambda\]
Where \(c\) is the constant speed of light, \(f\) is the frequency, and \(\lambda\) is the wavelength.

When you're solving for one of these variables, you rearrange the equation appropriately. For frequency, you isolate \(f\) by dividing both sides by \(\lambda\):

• Frequency formula: \(f = \frac{c}{\lambda}\)

To understand how these variables are related, consider a guitar string. As you tighten the string (increasing the frequency), the pitch (analogous to light's color) gets higher. For light, as the wavelength gets shorter (blue light), the frequency goes up; conversely, longer wavelengths (red light) have lower frequencies. This relationship is the same essence captured in the wave equation and applies to all types of waves.

Speed of Light

The speed of light, denoted by \(c\), is one of the most important constants in physics. It is the speed at which all electromagnetic radiation travels in a vacuum, and it is approximately \(3\times10^8 \, \mathrm{m/s}\). This incredible speed allows light to travel around the Earth roughly 7.5 times in just one second!

The speed of light is not just a fascinating trivia; it's crucial for understanding not only wave phenomena but also the nature of the universe. It forms the basis of Einstein's theory of relativity and affects how we perceive time and space.

In our context, the speed of light provides the link between frequency and wavelength. When performing calculations with the wave equation, you can rely on the constant nature of \(c\) to find unknown quantities. The unwavering speed of light ensures that even if we don't know the frequency or wavelength of light, determining one allows us to find the other, provided we have the wave equation in hand. This constant is why we could solve for the frequency of the CD player's laser light with such precision.