a) Write the mathematical equation that relates the energy of a photon and its wavelength. b) Is the energy of a photon proportional or inversely proportional to \(\lambda\) ? $$ \begin{array}{ll} \hline \text { Region } & \text { Wavelength Range } \\ \hline \text { radiowave } & 3 \mathrm{~km}-30 \mathrm{~cm} \\ \text { microwave } & 30 \mathrm{~cm}-1 \mathrm{~mm} \\ \text { infrared (IR) } & 1 \mathrm{~mm}-800 \mathrm{~nm} \\ \text { visible (VIS) } & 800 \mathrm{~nm}-400 \mathrm{~nm} \\ \text { ultraviolet (UV) } & 400 \mathrm{~nm}-10 \mathrm{~nm} \\ \text { X-ray } & 10 \mathrm{~nm}-0.1 \mathrm{~nm} \\ \text { gamma ray } & <0.1 \mathrm{~nm} \\ \hline \end{array} $$ $$ \begin{array}{ccc} \hline \hline \begin{array}{c} \text { Wavelength } \\ (\mathrm{nm}) \end{array} & \begin{array}{c} \text { Frequency } \\ \left(10^{14} \mathrm{~s}^{-1}\right) \end{array} & \begin{array}{c} \text { Energy } \\ \left(10^{-19} \mathrm{~J}\right) \end{array} \\ \hline 333.1 & 9.000 & 5.963 \\ 499.7 & 6.000 & 3.976 \\ 999.3 & 3.000 & 1.988 \\ \hline \end{array} $$

Let's delve into the concept of photon energy, which is a crucial part of understanding how light behaves on a particle level. A photon is a quantum of electromagnetic energy, essentially a packet of light. Its energy is dependent on its electromagnetic properties, namely its frequency, and is described by the Planck-Einstein relation. This fundamental relation is given by the formula:
\[\begin{equation} E = h \times c / \lambda \end{equation}\]
where \( E \) is the photon's energy, \( h \) is Planck’s constant (approximately \( 6.626 \times 10^{-34} \) joule seconds), \( c \) is the speed of light in a vacuum (around \( 3 \times 10^8 \) meters per second), and \( \lambda \) stands for the wavelength of the photon.

The energy of a photon is a direct measure of its ability to interact with other particles or fields. This interaction is the basis for a wide range of phenomena, from the generation of an electrical current in solar panels through the photoelectric effect to our very perception of light and color in our eyes. In the conceptual world of quantum mechanics, photons with higher energies can be seen as more 'powerful' in their effects when they interact with matter.

Moving onto the concept of wavelength proportionality, it's important to note that photon energy and wavelength are inversely related, as described by the Planck-Einstein relation presented above.
To make this concept more tangible, consider this: when you look at a rainbow, you are actually seeing light that has been dispersed into various wavelengths, which we perceive as different colors. Violet light has a shorter wavelength compared to red light. According to the inverse proportionality, violet light photons carry more energy than red light photons.

Understanding this relationship helps us in various scientific applications. For instance, in medical imaging, shorter wavelengths (like X-rays) are used because their higher photon energies can penetrate through the body, allowing us to see inside.

Another way to view this proportionality is through the formula:\[\begin{equation} \lambda \propto \frac{1}{E} \end{equation}\]
As \( \lambda \) increases, which means the wavelength is getting longer, the energy \( E \) decreases. This inverse relationship indicates that to increase the energy of a photon, one must decrease its wavelength.

The electromagnetic spectrum is the range of all types of electromagnetic radiation. Radiation is often categorized by wavelength, creating a spectrum that includes, in order of increasing energy and decreasing wavelength: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each category corresponds to a different segment of the wavelength range, as outlined in the exercise table.

This spectrum illustrates the vast diversity of electromagnetic waves and their varying applications. Radio waves, with their longest wavelengths, are used for communication such as broadcasting and satellite transmissions. Microwaves have enough energy to cause water molecules to heat, hence their use in cooking.

Infrared is often associated with thermal imaging, while visible light enables the sense of sight. Moving into the higher energy parts of the spectrum, ultraviolet radiation can lead to chemical reactions and is used in sterilization. X-rays can pass through various materials, aiding in medical diagnostics, and gamma rays have applications in cancer treatment due to their penetrating power.

Understanding the electromagnetic spectrum is not just about knowing the different types of radiation; it's about grasping the physical properties that govern their interactions with matter and their implications in diverse contexts from astronomy to medicine.