The student is expected to understand the electromagnetic spectrum and the mathematical relationships between energy, frequency, and the wavelength of light AND calculate the wavelength, frequency, and energy of light using Planck’s constant and the speed of light.
The electromagnetic spectrum displays the full range of electromagnetic energy based on wave properties, from high-energy gamma rays to low-energy radio waves. Electromagnetic waves are characterized by energy, frequency, and wavelength.
A wave equation mathematically describes the properties of a wave using velocity, frequency, and wavelength. All electromagnetic waves travel at the same velocity known as the speed of light (c), which equals 3.0 x 108 m/s. Wavelength (λ) is defined as the distance between two peaks or two troughs on a wave. Frequency (f) is defined as the number of waves passing a given point per second.
The mathematical relationship between the frequency and wavelength of a electromagnetic wave is given by the equation c = λ x f, where the speed of light equals the product of the frequency (f) and wavelength (λ). Both the frequency and wavelength of the wave may be calculated using the speed of light (c). Since the speed of light is a constant, frequency and wavelength are inversely proportional.
The energy of an electromagnetic wave (E) may be calculated using a physical constant known as Planck’s constant (h) and the frequency (f) of the wave in the formula E = hf. Planck’s constant equals 6.63 x 10-34 J x s. According to this equation, the energy of an electromagnetic wave is directly proportional to its frequency. The energy of electromagnetic wave may also be calculated using the speed of light (c), the wavelength of light (λ), and Planck’s constant (h) in the formula E = hc/λ.
The Electromagnetic Spectrum
The term spectrum originated from the Latin word that means image, appearance, or apparition. The electromagnetic spectrum displays the full range of electromagnetic energy based on wave properties, from high-energy gamma rays to low-energy radio waves.
Physical Properties of Electromagnetic Waves
Electromagnetic waves are characterized by wavelength, frequency, and energy.
Wavelength: Wavelength (λ) is defined as the distance between two peaks or two troughs on a wave. It is measured in meters. Visible light has wavelengths from 390 to about 700 nanometers (nm). The longest wavelengths are radio waves, which are about the length of a football field. The shortest wavelengths are gamma rays that have lengths measured at the atomic level.
Frequency: Frequency (f) is defined as the number of waves passing a given point per second. It is measured in a unit known as Hertz (hz). For the visible light spectrum, the frequency of the wave determines its color. For example, the wavelength 41014 Hz is red light, where 81014 Hz is violet light. Between these wavelengths are all the other colors of the rainbow. An electromagnetic wave can have a frequency less than 41014 Hz, but it will be invisible to the human eye. Such waves are called infrared (IR) radiation. At even lower frequencies, electromagnetic waves known as microwaves. At still lower frequencies, they are known as radio waves. An electromagnetic wave can have a frequency higher than 81014 Hz and will be invisible to the human eye as well. Such waves are called ultraviolet (UV) radiation. Waves with frequencies higher than this are called X-rays, and higher still are called gamma rays.
Energy: The energy carried by an electromagnetic wave is proportional to the frequency of the wave. Electromagnetic waves that are of higher energy than visible light (higher frequency, shorter wavelength) include ultraviolet light, X-rays, and gamma rays. Lower energy waves (lower frequency, longer wavelength) include infrared light, microwaves, and radio and television waves. The energy of visible light frequencies fall between those two extremes. Energy is measured in electron volts.
Wavelength and Frequency are Inversely Related
Compared to all other electromagnetic waves, radio waves have the lowest frequencies and longest wavelengths. From radio waves to the microwaves, wave frequencies increase and wavelengths decrease. This relationship continues through the spectrum toward gamma rays, which have the highest frequencies and shortest wavelengths. Thus, as wavelength increases, frequency decreases, and vice versa. This occurs as there is an inverse relationship between wavelength and frequency. One can write a mathematical equation to show this relationship as a wave equation. All energy waves travel at the same velocity known as the speed of light (c), which equals 3.0 x 108 m/s. Simply put,
Speed of light = (frequency) (wavelength)
The speed of light is represented by the letter c. Frequency is represented by the letter f. Wavelength is represented by the symbol lambda (λ). The mathematical relationship between the frequency and wavelength of an electromagnetic wave is given by the equation:
c = λ • f
In the formula above, the speed of light equals the product of the wavelength (λ) and frequency (f). Both the frequency and wavelength of the wave may be calculated using the speed of light (c). Since the speed of light is a constant, frequency and wavelength are inversely proportional. It is scientific convention to use 3.0 x 108 m/s is used as the speed for light. The actual value of 299,792,458 m/s. However, this value only holds for the speed of light in a vacuum, such as space. In other media, the speed of light can vary greatly. For example, the speed of light in glass is around 2.0 x 108 m/s.
Calculating Wave Energy
Light definitely has a wave-like nature, but it also has a particle-like nature. It is the particle nature of light that is responsible for the colors produced by the metal ions when held in a flame. If light were only made of waves, then the metals in the flame would emit all colors and wavelengths of light. Instead, we see only certain wavelengths emitted from each metal. These correspond to particles of light with energies specific to the metals that emit them. Max Planck performed tests on metals that emitted certain colors of light and named the particles of light photons. He determined a formula that relates the energy of a photon of light to its wavelength using a constant that bears his name. The energy of an electromagnetic wave (E) may be calculated using a physical constant known as Planck’s constant (h) and the frequency (f) of the wave in the formula:
Energy = (Planck’s constant) (frequency)
And it can be written as the following:
E = h • f
Planck’s constant equals 6.63 x 10-34 J•s. According to this equation, the energy of an electromagnetic wave is directly proportional to its frequency. From this equation, we know that the energy of electromagnetic waves may also be calculated using the speed of light (c), the wavelength of light (λ), and Planck’s constant (h). This equation yields the following:
f = c / λ
E = h • f = h • c / λ
This gives the following formula:
E = hc/ λ