Question

What does the wavelength of electromagnetic radiation represent? How is the wavelength of radiation related to the energy of the photons of the radiation?

Short answer

The wavelength of electromagnetic radiation represents the distance between two consecutive peaks or troughs of a wave and is usually denoted by \(\lambda\). Wavelength is important because it determines the frequency, energy, and type of electromagnetic wave. The energy of a photon is related to the wavelength using Planck's constant (\(h\)) and the speed of light (\(c\)). The energy of a photon is inversely proportional to its wavelength (\(E \propto \frac{1}{\lambda}\)), meaning that as the wavelength increases, the photon carries less energy, and vice versa.

Step-by-step solution

Definition of Wavelength

The wavelength of electromagnetic radiation represents the distance between two consecutive peaks or troughs of a wave. It is usually denoted by the symbol \(\lambda\) and is measured in meters. Wavelength is an important property of electromagnetic waves because it determines the frequency, energy, and even the type of electromagnetic wave (e.g., radio, X-rays, etc.).

Relationship between Wavelength, Frequency and Speed of Light

The wavelength of electromagnetic radiation is related to its frequency (\(f\)), which is the number of wave cycles (or oscillations) that occur per unit of time, typically measured in Hertz (Hz). The two are related through the speed of light (\(c\)), which is a constant value approximately equal to \(3.0 \times 10^8\) meters per second. The formula that links wavelength, frequency, and the speed of light is: \[c = \lambda \times f\] This equation tells us that the product of wavelength and frequency of an electromagnetic wave is equal to the speed of light. Consequently, if the wavelength of a wave increases, its frequency will decrease, and vice versa.

Relationship between Wavelength and Energy of Photons

Now let's discuss the relation between the wavelength of electromagnetic radiation and the energy of its photons. A photon is a particle of light that carries energy. The energy of a photon is related to the frequency of the radiation using Planck's constant (\(h\)), which is approximately equal to \(6.626 \times 10^{-34}\) Joule seconds (Js). The formula is: \[E = h \times f\] Since we know the relation between wavelength and frequency from the previous step, we can rewrite this equation in terms of wavelength by substituting the value of frequency from \(c = \lambda \times f\): \[f = \frac{c}{\lambda}\] Now, replace \(f\) with \(\frac{c}{\lambda}\) in the energy equation: \[E = h \times \frac{c}{\lambda}\] This equation shows that the energy of a photon is inversely proportional to its wavelength: \[E \propto \frac{1}{\lambda}\] Thus, when the wavelength of an electromagnetic wave is longer, the photons carry less energy, and when the wavelength is shorter, the photons have more energy. This relationship is essential in understanding the behavior and properties of different types of electromagnetic waves.

Key concepts

Wavelength and Frequency Relationship

Understanding the relationship between the wavelength and frequency of electromagnetic radiation is fundamental in physics and many technological applications. The wavelength (\texttt{\(\lambda\)}) is the physical distance between two identical points on a wave, such as peak to peak or trough to trough. Frequency (\texttt{\(f\)}), on the other hand, represents the number of waves that pass a given point per second.

How exactly are wavelength and frequency connected? They are inversely related to each other; as one increases, the other decreases. This relationship is due to the fact that all electromagnetic radiation moves through a vacuum at a constant speed, the speed of light, which leads us into the next key concept.

Speed of Light

The speed of light (\texttt{\(c\)}) is a universal physical constant that plays a pivotal role in the realm of physics. In a vacuum, light travels at an astonishingly fast pace of approximately \texttt{\(3.0 \times 10^8\)} meters per second. Whether we're discussing gamma rays, X-rays, ultraviolet light, visible light, infrared, microwaves, or radio waves, they all travel at this unchanging speed when in a vacuum.

It's the steadfast nature of the speed of light that anchors the wavelength and frequency relationship. Consider the formula \texttt{\(c = \lambda \times f\)}. Since \texttt{\(c\)} is constant, any change in the wavelength (\texttt{\(\lambda\)}) must be accompanied by an inverse change in the frequency (\texttt{\(f\)}), to ensure the equation remains balanced. This simple yet profound relationship is why we can tune into different frequencies on a radio to hear different stations—each frequency corresponds to a different wavelength of radio wave.

Photon Energy

Delving into the quantum world, photon energy is a crucial concept that links the wave nature of light with its particle characteristics. Each photon is a quantum, or packet, of energy associated with electromagnetic radiation. The energy (\texttt{\(E\)}) of a photon is directly proportional to its frequency (\texttt{\(f\)}) and can be calculated using Planck's constant (\texttt{\(h\)}), through the formula \texttt{\(E = h \times f\)}. Planck's constant, a fundamental value in quantum mechanics, is around \texttt{\(6.626 \times 10^{-34}\)} Joule seconds.

Thanks to the interconnected nature of frequency, wavelength, and the speed of light, this equation also reveals an inverse relationship between photon energy and wavelength. Mathematically, we can express this as \texttt{\(E \propto \frac{1}{\lambda}\)}, indicating that a shorter wavelength (and thus higher frequency) corresponds to a higher-energy photon. This is why ultraviolet light can cause sunburns whereas radio waves do not; ultraviolet has a much shorter wavelength compared to radio waves, meaning its photons pack more energy capable of affecting our skin cells.