Question

Calculate the energy in joules of a photon of green light having a wavelength of \(563 \mathrm{nm}\).

Short answer

The energy of a green light photon with a wavelength of 563 nm is approximately 3.527 x 10⁻¹⁹ J.

Step-by-step solution

Convert the wavelength from nanometers to meters

To convert the wavelength of the photon from nanometers (nm) to meters (m), divide the value by the conversion factor of 1 billion, since 1 nm = 10⁻⁹ m. Therefore, a wavelength of 563 nm is equal to 563 x 10⁻⁹ m.

Use the speed of light and Planck's constant

The energy (E) of a photon can be calculated using the equation E = hc/λ, where h is Planck's constant (h = 6.626 x 10⁻³⁴ J·s), c is the speed of light in a vacuum (c = 3.0 x 10⁸ m/s), and λ is the wavelength in meters.

Calculate the energy of the photon

Substitute the values into the equation: E = (6.626 x 10⁻³⁴ J·s) x (3.0 x 10⁸ m/s) / (563 x 10⁻⁹ m).

Perform the calculation

Multiply Planck's constant by the speed of light and then divide by the converted wavelength to get the energy of the photon in joules.

Simplify to find the answer

E = (6.626 x 10⁻³⁴ x 3.0 x 10⁸) / (563 x 10⁻⁹). After performing the calculation, you will get the energy E in joules.

Key concepts

Planck's Constant

Planck's constant is a fundamental value in the realm of quantum mechanics, symbolized by 'h'. This constant is a central part of one of the most important equations in quantum physics: the energy-wavelength equation for photons. Introduced by Max Planck, it describes the quantization of energy in the interaction between matter and electromagnetic radiation.

With a value of approximately \(6.626 \times 10^{-34}\) joule-seconds (J·s), Planck’s constant permits us to calculate the energy carried by a single photon given its frequency or, inversely, its wavelength. This calculation is foundational for understanding the quantum nature of light and illustrates the particle-like behavior of electromagnetic radiation, a concept known as the wave-particle duality. Understanding Plant's constant is crucial for anyone delving into studies of light, energy, and the intrinsic characteristics of the quantum world.

Speed of Light

The speed of light in a vacuum, represented by the symbol 'c', is another pivotal constant in physics, essential for calculations involving light and electromagnetic waves. It has a fixed value of \(3.0 \times 10^{8}\) meters per second (m/s) and is utilized to connect the properties of wavelength and frequency with energy.

This constant not only grounds the theory of special relativity put forth by Albert Einstein but also plays a significant role in computing the energy of photons. Since energy is inversely proportional to the wavelength, knowing the speed of light allows us to link spatial properties of light to its energy content, which is elemental in photon energy calculations. The concept that light always travels at this constant speed in a vacuum regardless of the observer's motion reveals profound implications about the nature of time and space.

Wavelength Conversion

When calculating photon energy, it's common to encounter wavelengths in different units, necessitating conversion for proper calculation. Most commonly, wavelengths are given in nanometers (nm) and must be converted to meters (m), the standard unit used in scientific equations.

To convert from nanometers to meters, we divide by \(1 \times 10^{9}\), since there are one billion nanometers in a meter. This conversion is crucial because mismatched units can lead to errors in calculations. By standardizing the unit of length to meters, we ensure consistency and accuracy when applying the energy-wavelength equation. The task of conversion isn't only a mathematical exercise but ensures clarity and precision in the realm of physics, where even the smallest discrepancy can lead to vastly different outcomes.

Energy-Wavelength Equation

The energy-wavelength equation is vital for understanding how energy is quantified for photons. The equation \( E = \frac{hc}{\lambda} \) shows the relationship between the energy (E) of a photon, Planck's constant (h), the speed of light (c), and the photon's wavelength (\(\lambda\)).

By applying this equation, one can calculate that a photon with a particular wavelength carries a specific amount of energy. The equation implies that the shorter the wavelength of light, the higher its energy because energy is inversely proportional to the wavelength. Therefore, blue light, with a shorter wavelength than red light, carries more energy per photon. This equation exemplifies a fundamental principle in quantum physics that energy, at a microscopic scale, comes in discrete amounts or 'quanta.' Understanding this equation is not only central to mastering concepts in physics but also in various practical applications like spectroscopy, laser technology, and even understanding the nature of sunlight.