Question

a. How are the wavelength and frequency of electromagnetic radiation related? b. How are the energy and frequency of electromagnetic radiation related? c. How are the energy and wavelength of electromagnetic radiation related?

Short answer

Wavelength and frequency are inversely related: \[ c = \lambda \u \]. Energy and frequency are directly related: \[ E = h \u \]. Energy and wavelength are inversely related: \[ E = \dfrac{hc}{\lambda} \].

Step-by-step solution

Understanding wavelength and frequency relationship

The wavelength \( \lambda \) and frequency \( \u \) of electromagnetic radiation are inversely related. The equation that describes this relationship is \[ c = \lambda \u \] where \( c \) is the speed of light (approximately \( 3.00 \times 10^8 \) meters per second).

Relating energy to frequency

The energy \( E \) of electromagnetic radiation is directly proportional to its frequency \( \u \). The relationship is given by the equation \[ E = h \u \] where \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \) joule seconds).

Connecting energy and wavelength

Because energy is related to frequency and frequency is related to wavelength, energy \( E \) and wavelength \( \lambda \) are also related. By combining the equations from Steps 1 and 2, we get \[ E = \dfrac{hc}{\lambda} \] which shows that energy is inversely proportional to wavelength.

Key concepts

wavelength and frequency relationship

First, let's look at the relationship between wavelength and frequency for electromagnetic radiation. Wavelength (symbolized as \( \lambda \)) and frequency (symbolized as \( \u \)) are inversely proportional to each other. This means that as one increases, the other decreases. The formula that describes this relationship is \[ c = \lambda \u \], where \( c \) is the speed of light, approximately \( 3.00 \times 10^8 \) meters per second.

Simply put, if the wavelength is longer, the frequency is lower. Conversely, if the wavelength is shorter, the frequency is higher.
For example, radio waves have long wavelengths and low frequencies, while gamma rays have very short wavelengths and high frequencies.

energy and frequency relationship

Next, let's understand how the energy of electromagnetic radiation relates to its frequency. Energy (denoted as \( E \)) is directly proportional to frequency \( \u \). This relationship is given by the equation \[ E = h \u \], where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \) joule-seconds.

This means that as the frequency of the electromagnetic radiation increases, so does its energy, and vice versa.
For instance, x-rays have a higher frequency and thus greater energy, compared to visible light, which has lower energy due to its lower frequency.

energy and wavelength relationship

Lastly, let's explore the relationship between the energy and wavelength of electromagnetic radiation. Since we know from previous sections that wavelength and frequency are inversely proportional and that energy is directly proportional to frequency, we can also relate energy to wavelength.
Using the equations \( c = \lambda \u \) and \( E = h \u \), we can combine them into one equation: \[ E = \dfrac{hc}{\lambda} \], which shows that energy is inversely proportional to wavelength.

In simple terms, if the wavelength is short, the energy is higher. Conversely, if the wavelength is long, the energy is lower.
This can be observed in phenomena like ultraviolet light having shorter wavelengths and higher energy compared to infrared light, which has longer wavelengths and lower energy.