To emit the same amount of light energy per second, which must emit more photons per second: a source of red light, or a source of blue light? Explain.

The concept of a photon's energy is pivotal to understanding electromagnetic radiation, which includes light. At a fundamental level, a photon is a particle representing a quantum of light, and it carries a specific amount of energy. This energy can be calculated using the equation \(E = hv\), where \(h\) symbolizes Planck's constant, and \(v\) represents the frequency of the photon. The key takeaway is that the energy contained within a photon is directly proportional to its frequency; higher frequency photons carry more energy.

For example, when comparing two different colors of light, say red and green, green light has a higher frequency than red light. Therefore, according to the equation mentioned above, green photons have more energy than red photons. This concept is crucial when examining how different colors of light interact with substances and how they contribute to various phenomena like the color of objects, or in this case, the number of photons needed to emit a certain amount of energy.

Planck’s constant (symbolized as \(h\)) is a fundamental figure in quantum mechanics, establishing a relationship between the energy of a photon and its frequency. The value of Planck's constant is approximately \(6.626 × 10^{-34} \) Joule-seconds. The importance of Planck’s constant stems from its role as a proportionality factor, making it possible to link the wave-like behavior of light to its particle-like properties. Because its value is so small, the energies of photons are correspondingly tiny, requiring many photons to be involved in exchanges of energy that are noticeable at the macroscopic scale.

Without Planck's constant, we wouldn't be able to quantify the behavior of photons typically observed only at microscopic scales, such as in the photoelectric effect, where light causes electrons to be ejected from a metal surface.

Light behaves both as a wave and as a particle, and when we focus on its wave-like properties, two key characteristics arise: frequency and wavelength. The frequency (often denoted by \(v\) or \(u\)) of a light wave refers to the number of wave cycles that pass a given point each second. Meanwhile, the wavelength (denoted by \(\lambda\)) is the physical distance between successive peaks of the wave. These two properties are inversely related; higher frequency correlates with shorter wavelength and vice versa.

This relationship is mathematically expressed by the equation \(v = c / \lambda\), where \(c\) is the speed of light. Understanding the inverse relationship between frequency and wavelength is vital to deciphering the energetics of different colors of light, as frequency directly ties into the photon's energy. Thus, light with a short wavelength, which is high in frequency, is more energetic than light with a long wavelength.

Red and blue light represent two distinct portions of the visible light spectrum, each with its own characteristic wavelength and frequency. Red light, known for its longer wavelength, sits at the lower end of the visible light spectrum and carries less energy per photon. In contrast, blue light has a much shorter wavelength, placing it towards the higher end of the spectrum, and thus it has more energetic photons.

When comparing sources of red and blue light that emit energy at an equal rate, the source emitting red light compensates for its lower energy photons by emitting a greater number of them. On the other hand, the source emitting blue light requires fewer photons to deliver the same total energy, due to each blue photon being more energetic. This systematic difference in photon emission is intrinsic to the fundamental energy properties as described by the electromagnetic spectrum.