Question

Calculate the frequency of light that has a wavelength of \(75.0 \mathrm{~nm}\). What type of radiation is this?

Short answer

The frequency of light with a wavelength of \(75.0 \mathrm{~nm}\) is approximately \(4.00 \times 10^{15}\) Hz. This light corresponds to ultraviolet radiation.

Step-by-step solution

Convert Wavelength

Before proceeding with the frequency calculation, first convert the wavelength from nanometers to meters. Since \(1 \mathrm{m} = 10^9 \mathrm{nm}\), therefore, \(75.0 \mathrm{nm} = 75.0 \times 10^{-9} \mathrm{m}\).

Calculate Frequency

The frequency \(f\) can be calculated using the speed of light equation \(f = \frac{c}{\lambda}\). Given that the speed of light \(c\) is always \(3.00 \times 10^{8} \mathrm{m/s}\), the frequency is therefore \(f = \frac{3.00 \times 10^{8} \mathrm{m/s}}{75.0 \times 10^{-9} \mathrm{m}}\).

Find the Type of Radiation

The frequency that we calculated falls within the ultraviolet range of the electromagnetic spectrum, which represents light that has a wavelength from \(10^{-8}\) to \(10^{-7}\) meters. Therefore, the radiation from the light with this frequency is ultraviolet radiation.

Key concepts

Wavelength to Frequency Conversion

In the world of physics, converting wavelength to frequency is a fundamental process. Every form of light, including visible and invisible types, has both a wavelength and a frequency. The wavelength is the distance between successive peaks of a wave, and frequency is how often these waves pass a point within a certain period of time. This conversion is crucial because it allows us to understand more about the type of light and its energy.
To convert a wavelength to a frequency, we use the formula:

• \( f = \frac{c}{\lambda} \)

where \( f \) is the frequency, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
In the provided exercise, we started with a wavelength of \(75.0 \text{ nm} \). First, it is essential to convert nanometers into meters because the standard unit for these calculations is meters. We know that \(1 \text{ m} = 10^9 \text{ nm} \), hence, \(75.0 \text{ nm} = 75.0 \times 10^{-9} \text{ m}\). With this conversion, you can now proceed to use the formula efficiently, simply by substituting the value of the wavelength and the constant speed of light.

Ultraviolet Radiation

Ultraviolet (UV) radiation is a type of electromagnetic radiation just beyond visible light. It falls within the spectrum between violet light and X-rays. Humans cannot see UV radiation, but many organisms can, which has particular biological implications.
UV radiation is classified into three categories:

• UVA (320–400 nm)
• UVB (290–320 nm)
• UVC (100–290 nm)

Each type has varying effects on the biological world and materials.
In this exercise, after converting the wavelength from nanometers to meters, the type of radiation can be identified using its frequency. The answer confirms that the wavelength of \(75.0 \text{ nm} \) falls into the ultraviolet category based on the frequency calculated. Studying UV radiation is crucial because it affects everything from human health to the durability of materials and environmental conditions.

Speed of Light

The speed of light is an essential constant in physics that represents how fast light travels through a vacuum. This speed is universally fixed at approximately \(3.00 \times 10^{8} \text{ m/s} \). It is a critical factor in calculations across physics, notably in formulas that involve wave properties, including our frequency and wavelength conversion.
The speed of light is used frequently in equations because it connects frequency and wavelength. This constant provides the bridge needed to explore the wider electromagnetic spectrum. For instance, when calculating the frequency of a light wave from its wavelength, we divide the speed of light by the wavelength:

• \( f = \frac{c}{\lambda} \)

This relationship shows how tightly bonded these properties are thanks to the speed of light, allowing scientists to predict behaviors of different kinds of light and radiation.
Understanding the speed of light is critical in enhancing our knowledge of light's behavior and applications. It forms the backbone of many technological advances, including telecommunications and even theories of relativity.