Question

By what properties is an electromagnetic wave characterized? How are these properties related to each other?

Short answer

Answer: The key properties of an electromagnetic wave are amplitude, frequency, wavelength, and velocity. These properties are interrelated through various mathematical relationships: frequency and wavelength are inversely proportional ($f={\displaystyle \frac{c}{\lambda }}$), energy is directly proportional to frequency ($E=hf$), and velocity is determined by frequency and wavelength ($v=f\lambda$). Understanding these relationships helps in comprehending the behavior and properties of electromagnetic waves.

Step-by-step solution

Identify the properties of an electromagnetic wave

An electromagnetic wave is characterized by its amplitude, frequency, wavelength, and velocity. The amplitude determines the strength or intensity of the wave, the frequency measures the number of oscillations per second (measured in Hertz), the wavelength is the distance between two consecutive points in the same phase of the wave, and the velocity is the speed at which an electromagnetic wave travels.

Explain the relationship between wavelength and frequency

Wavelength ($\lambda$) and frequency ($f$) are inversely related. As the frequency of an electromagnetic wave increases, its wavelength decreases, and vice versa. This relationship is mathematically represented as $f={\displaystyle \frac{c}{\lambda }}$, where $c$ is the speed of light, which is approximately $3.0\times {10}^8\text{m/s}$.

Explain the relationship between frequency and energy

The energy ($E$) of a photon in an electromagnetic wave is directly proportional to its frequency. This relationship is given by the equation $E=hf$, where $h$ is Planck's constant, approximately equal to $6.63\times {10}^{-34}\text{Js}$. As the frequency of the wave increases, so does the energy of the individual photons, which in turn affect the intensity or brightness of the wave.

Describe the relationship between velocity, frequency, and wavelength

The velocity ($v$) of an electromagnetic wave is determined by the frequency and wavelength of the wave. This relationship is given by the equation $v=f\lambda$. In a vacuum, all electromagnetic waves travel at the speed of light ($c$), and this relationship is always true. In other media, the speed of light may vary slightly, depending on the refractive index of the medium, but the relationship between frequency and wavelength is still valid.

Conclusion

Electromagnetic waves are characterized by their amplitude, frequency, wavelength, and velocity. These properties are related to each other via a number of mathematical relationships, including $f={\displaystyle \frac{c}{\lambda }}$, $E=hf$, and $v=f\lambda$. Understanding these relationships is key to understanding the behavior and properties of electromagnetic waves.

Key concepts

Wavelength and Frequency Relationship

When diving into the world of electromagnetic waves, envisioning the ocean might be helpful. Just as the ocean's waves have a distance between their peaks (wavelength) and the number of peaks passing a point per second (frequency), so do electromagnetic waves. Electromagnetic waves travel through space at the speed of light, and the distance from one wave crest to the next is known as the wavelength, denoted by the Greek letter lambda $\lambda$. Conversely, frequency, denoted by the symbol $f$, refers to how many wave crests pass a certain point within a second.

The intriguing aspect is that wavelength and frequency are inversely related for electromagnetic waves. This is comparable to a guitar string; when you pluck it tighter, the pitch (frequency) increases while the waves along the string (wavelength) become shorter. In math terms, their relationship is denoted by the equation $f=\frac{c}{\lambda }$, where $c$ is the speed of light. So, if you know the speed of light and either the wavelength or frequency, you can calculate the other. For instance, if a radio wave has a lower frequency compared to a gamma-ray, you can expect that its wavelength is correspondingly longer.

Energy and Frequency Relationship

Moving on to the energy of electromagnetic waves, which gives them their 'umph,' as you might say in a casual conversation. This energy is crucial for things like heating your food in a microwave or getting a suntan. Now, let’s think of the energy in these waves as being carried by tiny packets, or photons. The amount of energy a photon carries is tied directly to the wave's frequency: the higher the frequency, the more energy each photon has.

The mathematical relationship that describes how energy is tied to frequency comes from the famous physicist Max Planck, represented by $E=hf$, where $E$ is energy, $f$ is frequency, and $h$ is Planck's constant. Without getting bogged down by the tiny numbers, just remember Planck's constant is a very small number that scientists have figured out after lots of experiments and math. So, a microwave emitting at a lower frequency has less energetic photons compared to the high-frequency photons zipping out of your remote controller's infrared signal.

Wave Velocity

Lastly, let's tackle wave velocity, the measure of how fast these electromagnetic ripples are traversing through space. It might come as a surprise, but all electromagnetic waves - whether they're the ones bringing music to your radio or an X-ray scanning your broken arm - travel at the same speed in a vacuum. That speed is what we call the speed of light, a snappy $3.0\times {10}^8$ meters per second, often represented by $c$.

In a vacuum, the formula $v=f\lambda$ shows off the relationship between velocity ($v$), frequency ($f$), and wavelength ($\lambda$). However, when these waves enter different materials like water, glass, or air, they might slow down a bit because they're interacting with the atoms and molecules in those substances. This change in velocity can affect how we perceive light, like how a straw looks bent in a glass of water. The phenomenon is tied to the concept of refractive index of the material, yet, no matter the medium, the core relationship between velocity, frequency, and wavelength remains solid.