Problem 3 - University Physics with Modern Physics

Question

A photon of green light has a wavelength of 520 nm. Find the photon's frequency, magnitude of momentum, and energy.Express the energy in both joules and electron volts.

Step-by-Step Solution

Answer

Frequency: \(5.77 \times 10^{14}\, \text{Hz}\), Energy: \(3.82 \times 10^{-19}\, \text{J}\) or \(2.38\, \text{eV}\), Momentum: \(1.27 \times 10^{-27}\, \text{kg} \cdot \text{m/s}\).

Step 1: Convert Wavelength to Meters

The wavelength of the photon is given as 520 nm. To perform calculations, convert this to meters by using the conversion: \[ \text{1 nm} = 10^{-9} \, \text{m} \]Thus, the wavelength \( \lambda \) in meters is:\[ \lambda = 520 \, \text{nm} = 520 \times 10^{-9} \, \text{m} = 5.2 \times 10^{-7} \, \text{m} \]

Step 2: Calculate the Frequency

Use the speed of light formula to find the frequency (\( u \)): \[ c = \lambda u \]where \( c = 3 \times 10^8 \, \text{m/s} \). Solving for frequency gives:\[ u = \frac{c}{\lambda} = \frac{3 \times 10^8}{5.2 \times 10^{-7}} = 5.77 \times 10^{14} \, \text{Hz} \]

Step 3: Calculate the Energy in Joules

The energy of a photon is given by Planck's equation: \[ E = h u \]where \( h = 6.626 \times 10^{-34} \, \text{Js} \) is Planck's constant. Substituting the frequency:\[ E = 6.626 \times 10^{-34} \times 5.77 \times 10^{14} = 3.82 \times 10^{-19} \, \text{J} \]

Step 4: Convert Energy to Electron Volts

To convert energy from joules to electron volts, use the conversion:\[ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \]Thus, the energy in electron volts is:\[ E = \frac{3.82 \times 10^{-19}}{1.602 \times 10^{-19}} = 2.38 \, \text{eV} \]

Step 5: Calculate the Momentum

The momentum (\( p \)) of the photon can be calculated using:\[ p = \frac{E}{c} \]Substituting the energy in joules and the speed of light:\[ p = \frac{3.82 \times 10^{-19}}{3 \times 10^8} = 1.27 \times 10^{-27} \, \text{kg} \cdot \text{m/s} \]

Key Concepts

Wavelength Conversion

Wavelength conversion is an essential step in photon energy calculations, especially when dealing with nanometers (nm), a common unit for measuring wavelengths of light. To convert nanometers to meters, which is the standard unit in physics, use the relation:

• 1 nanometer (nm) = 10^-9 meters (m)

This conversion is crucial because physical equations like those involving speed of light or energy require wavelengths to be in meters. For instance, a wavelength of 520 nm becomes 520 × 10^-9 meters, resulting in 5.2 × 10^-7 meters. This conversion helps maintain consistency across calculations.

Frequency Calculation

Finding the frequency of a photon involves using the basic relationship between wavelength and frequency dictated by the speed of light. The speed of light (\[ c \]) is a constant, approximately 3 × 10⁸ meters per second. The formula connecting wavelength (\[ \lambda \]), frequency (\[ u \]), and speed of light is:

• \[ c = \lambda u \]

To solve for frequency, rearrange the equation to:

• \[ u = \frac{c}{\lambda} \]

Substitute the given values to calculate frequency. For our 520 nm wavelength (5.2 × 10^-7 m), the calculation would be:

• \[ u = \frac{3 \times 10^8}{5.2 \times 10^{-7}} = 5.77 \times 10^{14} \text{ Hz} \]

This frequency tells us how many wave peaks pass a point in one second. Understanding this concept helps in various fields, including communications and optics.

Momentum of Photon

Photon momentum seems counterintuitive because photons are massless particles. However, they do carry momentum, evident through interactions like hitting surfaces and transmitting energy. The momentum (\[ p \]) of a photon is derived from its energy divided by the speed of light:

• \[ p = \frac{E}{c} \]

Using Planck's constant, we calculate photon energy (\[ E \]) with the equation \[ E = hu \] (where \[ h \] is Planck's constant). Given that the energy has already been calculated as 3.82 × 10^-19 J, divide by the speed of light:

• \[ p = \frac{3.82 \times 10^{-19}}{3 \times 10^8} = 1.27 \times 10^{-27} \text{ kg} \cdot \text{m/s} \]

This value indicates the momentum associated with the photon, important in understanding light's interaction with matter, such as exerting pressure on a solar sail.

Energy Conversion to Electron Volts

Energy conversion to electron volts (eV) is particularly useful in physics and engineering, where energies involved are often very small when expressed in joules. One electron volt is the amount of energy a single electron gains moving across an electric potential difference of one volt. The conversion factor is:

• 1 eV = 1.602 × 10^-19 joules (J)

To convert our calculated photon energy from joules (3.82 × 10^-19 J) into electron volts, use:

• \[ E = \frac{3.82 \times 10^{-19}}{1.602 \times 10^{-19}} = 2.38 \text{ eV} \]

This conversion is practical, making it easier to handle numbers when discussing energy levels in atomic physics or semiconductor devices.