Question

A photon has a wavelength of \(705 \mathrm{nm} .\) Calculate the energy of the photon in joules.

Short answer

The energy of the photon is approximately \(2.818 \times 10^{-19}\) joules.

Step-by-step solution

Convert Wavelength to Meters

The wavelength of the photon is given as 705 nm. First, we need to convert this into meters because the SI unit for length is meters. To do this, recall that 1 nm equals \(10^{-9}\) meters.Calculating, we get:\[705 \text{ nm} = 705 \times 10^{-9} \text{ m} = 7.05 \times 10^{-7} \text{ m}\]

Use the Energy Formula

The energy of a photon can be calculated using the formula:\[ E = \frac{hc}{\lambda} \]where:- \( E \) is the energy,- \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \),- \( c \) is the speed of light \( (3.0 \times 10^8 \text{ m/s}) \),- \( \lambda \) is the wavelength of the photon.

Substitute Values and Calculate Energy

Substitute the values into the formula:\[ E = \frac{6.626 \times 10^{-34} \text{ Js} \times 3.0 \times 10^8 \text{ m/s}}{7.05 \times 10^{-7} \text{ m}} \]Calculate \( E \):\[ E = \frac{1.9878 \times 10^{-25}}{7.05 \times 10^{-7}} \]\[ E \approx 2.818 \times 10^{-19} \text{ J} \]

Key concepts

Wavelength Conversion

Understanding how to convert the wavelength of a photon from nanometers to meters is crucial for solving problems in physics. Wavelength is often measured in nanometers (nm) because it corresponds to the size of the particles that interact with electromagnetic waves, like light. However, calculations for photon energy require that wavelength be expressed in meters, the standard SI unit for length.

Conversion is simple once you remember that 1 nanometer equals \(10^{-9}\) meters. Thus, to convert a wavelength given in nanometers, you multiply it by \(10^{-9}\) to change it into meters. For example, a wavelength of 705 nm can be expressed in meters as \(705 \times 10^{-9} = 7.05 \times 10^{-7}\) meters. This conversion is a fundamental step in calculating photon energy.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics that describes the relationship between the energy of a photon and its frequency. It is denoted by the symbol \(h\) and has a value of \(6.626 \times 10^{-34}\) Js (joule seconds). This constant is incredibly small, reflecting the quantum scale at which photons and other particles interact.

Named after Max Planck, who introduced it in the early 20th century, Planck's constant underlies many quantum phenomena and is indispensable in calculating photon energy. By incorporating this constant in the photon energy formula, we can relate the energy of a photon to its wavelength, illustrating the wave-particle duality of light.

Speed of Light

The speed of light, denoted as \(c\), is a fundamental constant of nature, essential for calculations involving electromagnetic waves. It has a fixed value of \(3.0 \times 10^8\) m/s (meters per second). This speed is constant in a vacuum, and light travels at this speed across large cosmic distances.

In terms of photon energy calculations, the speed of light links the frequency of a photon to its wavelength. Together with Planck's constant, it helps determine the energy of photons of different wavelengths. When using the formula \(E = \frac{hc}{\lambda}\), the speed of light ensures that energy calculations are accurate and consistent with physical laws.

Photon Energy Formula

The photon energy formula is a critical mathematical tool for determining the energy stored in a photon. This formula is expressed as \(E = \frac{hc}{\lambda}\), where \(E\) represents the energy of the photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the photon.

This equation is fundamental in the field of quantum mechanics and electromagnetic wave theory. By substituting the values for Planck’s constant and the speed of light into this formula, and converting the wavelength to meters, you can calculate the energy of any photon.

For example, by substituting the given values into the formula for a photon with a wavelength of 705 nm (converted to \(7.05 \times 10^{-7}\) meters), we find that the energy \(E\) calculates to approximately \(2.818 \times 10^{-19}\) J (joules). This energy calculation is essential for understanding how photons interact with matter.