Find energy of each of the photons which : (a) correspond to light of frequency \(3 \times 10^{15} \mathrm{~Hz}\) (b) have wavelength of \(0.50 \mathrm{~A}\).

A fundamental aspect of quantum mechanics, Planck's constant serves as a cornerstone in the calculation of photon energy. Named after Max Planck, the constant (\( h \)) has a value of approximately \( 6.626 \times 10^{-34} \text{Joule seconds} (J \times s) \) and plays a pivotal role in the relationship between energy and frequency for a photon. When calculating the energy of a photon, the formula we use is \( E = h \times f \) where \( E \) is the energy in Joules, \( h \) is Planck's constant, and \( f \) is the frequency in Hertz (Hz).

This constant is not just a number; it represents the quantization of energy levels. In other words, energy is not continuous but occurs in discrete 'packets' called quanta. A photon is a quantum of electromagnetic radiation and its energy is directly proportional to its frequency, which brings us to the concept of frequency of light.

The frequency of light (\( f \)) is an essential aspect when determining a photon's energy. Frequency refers to the number of wave cycles that pass a fixed point in one second, and it's measured in Hertz (\( Hz \) or cycles per second). High-frequency light, such as ultraviolet rays, has more energy than low-frequency light, such as infrared.

To illustrate, in the context of the given exercise, the frequency provided for part (a) is \( 3 \times 10^{15} \) Hz. Applying this value into the energy formula alongside Planck's constant, you can determine that a photon of this frequency carries significant energy. Understanding the frequency allows us to unlock the energy stored in a single photon of light. The direct proportionality between energy and frequency means that as one increases, so does the other.

Delving deeper into the properties of light, we encounter the intrinsic connection between wavelength (\(\text{denoted as } \( \) \( \(lambda\) \)\)) and frequency. These two properties of a wave are inversely related through the speed of light (\( c \)), expressed in the fundamental equation \( c = \( \(lambda\) \) \times f \) where \( c = 3 \times 10^8 m/s \) - the constant speed of light in a vacuum.

For instance, a photon with a lower wavelength has a higher frequency, leading to greater energy. Conversely, a greater wavelength implies a lower frequency and therefore less energy. This is why different wavelengths of light (such as red versus violet) have different energies, which we perceive as colors. In part (b) of the exercise, once you've determined the wavelength, you can rearrange this equation to solve for frequency, allowing you to use the energy formula covered earlier. The intimate relationship between wavelength and frequency highlights the beautiful harmony within the electromagnetic spectrum and is key to understanding phenomena across physics.