Question

The retina of a human eye can detect light when radiant energy incident on it is at least \(4.0 \times 10^{-17} \mathrm{~J}\). For light of 585 -nm wavelength, how many photons does this energy correspond to?

Short answer

Approximately 118 photons.

Step-by-step solution

Write the Energy-Photon Relationship Formula

The energy of one photon can be calculated using the formula \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of a photon, \( h \) is Planck's constant \( (6.63 \times 10^{-34} \text{ J s}) \), \( c \) is the speed of light \( (3.00 \times 10^{8} \text{ m/s}) \), and \( \lambda \) is the wavelength.

Convert Wavelength to Meters

Convert the given wavelength from nanometers to meters: \( 585 \text{ nm} = 585 \times 10^{-9} \text{ m} \).

Calculate Energy of a Single Photon

Use the energy formula to find the energy of one photon: \[ E = \frac{(6.63 \times 10^{-34} \, \text{J s}) \times (3.00 \times 10^{8} \, \text{m/s})}{585 \times 10^{-9} \, \text{m}} \approx 3.40 \times 10^{-19} \text{ J} \] per photon.

Find Number of Photons

To find the number of photons, divide the total energy by the energy per photon: \[ n = \frac{4.0 \times 10^{-17} \, \text{J}}{3.40 \times 10^{-19} \, \text{J/photon}} \approx 118 \] photons.

Key concepts

Planck's Constant

Planck's constant is a crucial quantity in quantum mechanics, symbolized by the letter "h".
This fundamental concept shows the relationship between energy and frequency for particles like photons.
Planck's constant (denoted as \(h\)) has a value of \(6.63 \times 10^{-34} \text{ J s}\).

• It's used to express the size of quanta or "packets" of energy.
• It's essential for calculating the energy of a photon.

In an equation, like the one used to find a photon's energy \(E = \frac{hc}{\lambda}\), Planck's constant interacts with other constants to determine energy.
Each photon's energy is calculated by multiplying Planck's constant by the speed of light, and then dividing by the wavelength.
This equation is central to many calculations in quantum physics, helping us understand how light and its particles behave.

Speed of Light

The speed of light, represented by \(c\), is a key constant in physics.
It plays a vital role in the formulas used to calculate photon energy.
The speed of light is precisely \(3.00 \times 10^{8} \text{ m/s}\).

• Light travels at this ultimate speed in a vacuum.
• This value becomes a part of calculations that solve for crucial information about photons.

When we calculate the energy of a photon, the speed of light is multiplied by Planck's constant in the formula \(E = \frac{hc}{\lambda}\).
This shows that light's speed is directly related to photon energy, impacting how we interpret light in both theoretical and practical applications.

Wavelength Conversion

Wavelength conversion is an important aspect to understand when dealing with calculations involving light.
The wavelength is the distance between successive crests of a wave and is commonly measured in nanometers (nm).

• 1 nanometer equals \(10^{-9}\) meters.
• Converting wavelength to meters is essential for calculations within SI units.

In the problem we explored, the wavelength of light given was 585 nm, which needed conversion to meters:
\(585 \text{ nm} = 585 \times 10^{-9} \text{ m}\).
This conversion allows us to apply it in the energy formula, \(E = \frac{hc}{\lambda}\), effectively and without error.
A proper understanding of wavelength conversion is key to performing accurate scientific calculations across various areas of physics.