Visible light having a wavelength of \(6 \times 10^{-7}\) \(\mathrm{m}\) appears orange. Compute the frequency. and energy of a photon of this light.

In the realm of physics, light is understood as an electromagnetic wave, and the color we perceive is a result of the light's frequency. The visible light frequency falls within a specific range of the electromagnetic spectrum, perceived by the human eye as different colors, from violet to red. For instance, orange light, as mentioned in the exercise, typically has a wavelength of around 600 nanometers (nm), which in scientific terms is expressed as a wavelength of around (6 x 10^-7) meters (m).

The frequency (f) of light is directly related to its wavelength ( (lambda)), and this relationship is characterized by the simple yet powerful wave equation (c = lambda * f), where (c) is the speed of light in a vacuum, roughly (3 x 10^8) meters per second (m/s). By rearranging this equation to solve for frequency, we get (f = c /lambda). Thus, knowing the wavelength of our orange light, we can calculate its frequency. This concept is crucial because frequency is directly related to energy, which leads us to the next fundamental in physics, Planck's constant.

Planck's constant (h) is a pivotal figure in the quantum mechanics world, representing the quantization of energy. It was introduced by Max Planck and has a value of approximately (6.63 x 10^-34) joule seconds (Js). This seemingly inconspicuous number is monumental because it bridges the gap between the classical physics interpretation of energy and the quantum mechanical model.

According to Planck's theory, energy is not continuous but comes in discrete 'packets' called quanta or photons in the context of light. The energy (E) of a single photon is directly proportional to its frequency (f), with Planck's constant being the proportionality constant, as shown in the equation (E = h * f). This means that each color of light, corresponding to a specific frequency, has photons with distinctive energies, defined by this constant.

The Quantum Leap in Energy Comprehension

By incorporating Planck's constant into our calculations, we can determine the precise energy of a photon of our orange light, shedding light on the quantized nature of the electromagnetic spectrum. It's an exercise not just in mathematics but in understanding the fundamental behavior of light at the quantum level.

The wave equation, a core component of classical physics, relates the speed of a wave ( (c)) to its frequency ( ( f)) and wavelength ( (lambda )). It's succinctly expressed as (c = lambda * f). For electromagnetic waves, and light in particular, this equation simplifies our understanding of how energy is conveyed through space.

The speed of light ( (c)), at approximately (3 x 10^8) m/s, is a constant in a vacuum, a cornerstone of Einstein's theory of relativity. In contrast, the frequency and wavelength are inversely proportional: as the frequency increases, the wavelength decreases, and vice versa. This balance ensures that the product of the two, representing the speed of light, remains unchanged.

Decoding the Speed and Frequency

By mastering the wave equation, we can decode various properties of light, such as its speed, frequency, and therefore its energy. When applying this equation to calculate the frequency in our exercise, it bridges the observed wavelength and the theoretical speed, leading to an understanding of the inherent energy within a photon.