Problem 1 - 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: \(5.77 \times 10^{14}\) Hz, Energy: \(3.82 \times 10^{-19}\) J or 2.39 eV, Momentum: \(1.27 \times 10^{-27}\) kg⋅m/s.

Step 1: Identify Given Values

We are given that the wavelength of green light \( \lambda = 520 \text{ nm} \). Remember to convert this to meters for calculations: \( \lambda = 520 \times 10^{-9} \text{ m} \).

Step 2: Use the Speed of Light Formula to Find Frequency

The formula to relate speed of light \( c \), wavelength \( \lambda \), and frequency \( f \) is \( c = f \cdot \lambda \). Here, the speed of light \( c = 3.00 \times 10^8 \text{ m/s} \). Solve for frequency \( f \):\[ f = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \text{ m/s}}{520 \times 10^{-9} \text{ m}} = 5.77 \times 10^{14} \text{ Hz} \]

Step 3: Calculate the Energy of the Photon in Joules

Use Planck's equation \( E = h \cdot f \), where Planck's constant \( h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \). Insert the frequency from Step 2:\[ E = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \times 5.77 \times 10^{14} \text{ Hz} = 3.82 \times 10^{-19} \text{ J} \]

Step 4: Convert the Energy from Joules to Electron Volts

1 electron volt \( (\text{eV}) \) is equivalent to \( 1.60 \times 10^{-19} \text{ J} \). Convert the energy calculated in Step 3:\[ E = \frac{3.82 \times 10^{-19} \text{ J}}{1.60 \times 10^{-19} \text{ J/eV}} = 2.39 \text{ eV} \]

Step 5: Calculate the Photon’s Momentum

Use the momentum formula \( p = \frac{h}{\lambda} \), where \( h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \). Insert the values:\[ p = \frac{6.63 \times 10^{-34} \text{ J} \cdot \text{s}}{520 \times 10^{-9} \text{ m}} = 1.27 \times 10^{-27} \text{ kg} \cdot \text{m/s} \]

Key Concepts

Wavelength

In physics, wavelength is the distance between consecutive crests or troughs of a wave. For light or any electromagnetic wave moving through a vacuum, the speed is constant, and that's the speed of light, denoted as \( c \). For visible light such as green light, which this exercise deals with, the wavelength is typically measured in nanometers (nm), which is a billionth of a meter.

• Green light has a wavelength of about 520 nm, which is 520 x 10^-9 meters when converted to standard scientific units.


• Wavelength is inversely proportional to frequency, meaning the longer the wavelength, the lower the frequency.

A key formula involving wavelength, frequency \( f \), and the speed of light \( c \) is \( c = f \cdot \lambda \). By rearranging this formula, you can find the frequency if the wavelength is known, as was done in the original solution.

Frequency

Frequency \( f \) is a measure of how often the waves of light (or any electromagnetic radiation) pass a point in space per second. It is measured in Hertz (Hz), with one Hertz equivalent to one cycle per second.

• For the photon of green light with a wavelength of 520 nm, the frequency was calculated using the formula \( f = \frac{c}{\lambda} \).


• The speed of light \( c \) is approximately \( 3.00 \times 10^8 \text{ m/s} \), allowing us to calculate the frequency as \( 5.77 \times 10^{14} \text{ Hz} \).

It's important to understand that frequency is part of what defines the energy of a photon. According to Planck's equation \( E = h \cdot f \), where \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \)), the energy is directly proportional to the frequency.

Momentum

Momentum is a concept typically used in contexts involving motion, mass, and speed. But for photons, which are massless particles of light, momentum is defined using a different approach. The momentum \( p \) of a photon can be calculated using the formula \( p = \frac{h}{\lambda} \), where \( h \) is Planck's constant and \( \lambda \) is the wavelength of the photon.

• This equation shows that even though photons have no mass, they possess momentum because they have energy.


• Thus, the greener light photons, with a wavelength of 520 nm (\( 520 \times 10^{-9} \text{ m} \)), has a momentum of \( 1.27 \times 10^{-27} \text{ kg} \cdot \text{m/s} \).

Considering the small magnitude of a photon's momentum, it illustrates why light, despite being powerful, doesn't exert any perceivable push when it hits objects in our day-to-day experiences.