Question

Which wavelength has the higher frequency: \(550 \mathrm{nm}\) or \(650 \mathrm{nm} ?\)

Short answer

Wavelength 550 nm has the higher frequency.

Step-by-step solution

Understanding the Relationship Between Wavelength and Frequency

The frequency (\( f \)) of a wave is inversely related to its wavelength (\( \lambda \)) by the equation:\[f = \frac{c}{\lambda}\]where \( c \) is the speed of light, approximately \(3.00 \times 10^8 \, \text{m/s}\). This means that a shorter wavelength corresponds to a higher frequency.

Convert Wavelengths into Meters

Wavelengths are often given in nanometers, but we need them in meters to use the speed of light given in m/s.- Convert \(550 \, \text{nm}\) to meters: \(550 \, \text{nm} = 550 \times 10^{-9} \, \text{m}\).- Convert \(650 \, \text{nm}\) to meters: \(650 \, \text{nm} = 650 \times 10^{-9} \, \text{m}\).

Calculate the Frequency for Each Wavelength

Using \( f = \frac{c}{\lambda} \) and \( c = 3.00 \times 10^8 \text{ m/s} \):- For \(550 \, \text{nm} = 550 \times 10^{-9} \, \text{m}\):\[f_1 = \frac{3.00 \times 10^8}{550 \times 10^{-9}} = 5.45 \times 10^{14} \, \text{Hz}\]- For \(650 \, \text{nm} = 650 \times 10^{-9} \, \text{m}\):\[f_2 = \frac{3.00 \times 10^8}{650 \times 10^{-9}} = 4.62 \times 10^{14} \, \text{Hz}\]

Compare the Frequencies

Compare the frequencies calculated:- Frequency for \(550 \, \text{nm}\): \(5.45 \times 10^{14} \, \text{Hz}\)- Frequency for \(650 \, \text{nm}\): \(4.62 \times 10^{14} \, \text{Hz}\)The frequency corresponding to \(550 \, \text{nm}\) is higher.

Key concepts

Speed of light

When we think about light traveling through space, one crucial thing we need to consider is the speed at which it moves. This speed is what we call the "speed of light." It is a constant value, which means it doesn't change no matter where or how we measure it, as long as we're in a vacuum. The speed of light is approximately \(3.00 \times 10^8\, \text{m/s}\).

Why does the speed of light matter? Well, it lets us link two important characteristics of waves: wavelength and frequency. By knowing the speed of light, we can easily determine one if we know the other. The relationship is given by the formula:

• \( c = \lambda \times f \)

where:

• \( c \) is the speed of light,
• \( \lambda \) is the wavelength in meters, and
• \( f \) is the frequency in hertz.

It's fascinating how this simple relationship allows us to understand more about light and various other electromagnetic waves.

Nanometers to meters conversion

In science, especially when working with light, you often need to convert measurements from one unit to another. Wavelengths can be given in nanometers (nm), a very tiny measurement perfect for small distances like the wavelength of visible light. However, calculations involving the speed of light require us to use meters.

Here’s how you convert nanometers to meters. Remember this handy conversion relation:

• 1 nanometer (nm) = \(10^{-9}\) meters (m).

Let’s take an example to understand better:

• To convert \(550 \text{ nm}\) to meters, multiply by \(10^{-9}\):
  • \(550 \text{ nm} = 550 \times 10^{-9} \text{ m} = 5.50 \times 10^{-7} \text{ m} \).
• Similarly, \(650 \text{ nm}\) is converted to meters as:
  • \(650 \text{ nm} = 650 \times 10^{-9} \text{ m} = 6.50 \times 10^{-7} \text{ m} \).

By converting these units, you ensure your wavelength fits perfectly with other scientific calculations involving meters, like those using light’s speed.

Frequency calculation

Once you've converted the wavelength from nanometers to meters, you can use that information to find the frequency of the light wave. Frequency tells us how many wave cycles pass a given point per second and is measured in hertz (Hz). The formula you use is:

• \( f = \frac{c}{\lambda} \)

This means you divide the speed of light \( (c = 3.00 \times 10^8 \text{ m/s}) \) by the wavelength \( (\lambda \text{ in meters}) \).

Here's how you calculate frequency for specific wavelengths:

• For \(550 \text{ nm}\), already converted to meters as \(5.50 \times 10^{-7} \text{ m}\), the frequency \( (f_1) \) is:
  • \( f_1 = \frac{3.00 \times 10^8}{5.50 \times 10^{-7}} = 5.45 \times 10^{14} \text{ Hz} \)
• For \(650 \text{ nm}\), converted to \(6.50 \times 10^{-7} \text{ m}\), its frequency \( (f_2) \) is:
  • \( f_2 = \frac{3.00 \times 10^8}{6.50 \times 10^{-7}} = 4.62 \times 10^{14} \text{ Hz} \)

From these calculations, you can see that the shorter the wavelength, the higher its frequency. This understanding helps in analyzing different types of electromagnetic waves, from radio waves to X-rays.