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 575 -nm wavelength, how many photons does this correspond to?

Short answer

116 photons.

Step-by-step solution

Convert Wavelength to Meters

To solve this problem, we first need to convert the given wavelength from nanometers to meters. The wavelength is given as 575 nm, which is equivalent to:\[ 575 \text{ nm} = 575 \times 10^{-9} \text{ m} \]

Calculate the Energy of One Photon

We use the formula to calculate the energy of a single photon:\[ E = \frac{hc}{\lambda} \]where \( h = 6.626 \times 10^{-34} \text{ Js} \) is Planck's constant, and \( c = 3.00 \times 10^8 \text{ m/s} \) is the speed of light. Substituting the values:\[ E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{575 \times 10^{-9}} \]Calculating this gives:\[ E = 3.45 \times 10^{-19} \text{ J} \per \text{photon} \]

Calculate Number of Photons

To find out the number of photons required for the given radiant energy, we simply divide the total energy by the energy of one photon:\[ \text{Number of photons} = \frac{4.0 \times 10^{-17} \text{ J}}{3.45 \times 10^{-19} \text{ J/photon}} \]Calculating this value gives:\[ \text{Number of photons} = 116 \]

Key concepts

Wavelength Conversion

When dealing with wavelengths, it's common to find them in units like nanometers (nm).
This is because nanometers are convenient for measuring lengths at the atomic and molecular scale. However, in physics calculations, wavelengths are often required in meters for consistency with other units.

• 1 nanometer (nm) is equal to \( 10^{-9} \) meters (m).
• Converting a wavelength from nanometers to meters is straightforward: multiply the value by \( 10^{-9} \).

For example, to convert 575 nm to meters, you multiply it by \( 10^{-9} \):\[ 575\, \text{nm} = 575 \times 10^{-9} \text{m} \]This ensures that your wavelength is in the standard unit of meters, ready for use in further calculations.

Planck's Constant

Planck's constant is a fundamental constant in physics, symbolized by \( h \).
It relates the energy of a photon to its frequency. Max Planck, a German physicist, introduced this constant while solving the black-body radiation problem.

• Planck's constant \( h \) is approximately \( 6.626 \times 10^{-34} \) joule-seconds (Js).
• The energy \( E \) of a photon is given by \( E = h u \), where \( u \) is the frequency of the photon.

The value of Planck's constant is crucial in calculations involving quantum mechanics and explains how light energy is quantized in discrete packets known as photons.

Speed of Light

The speed of light is a universal constant denoted by \( c \).
It represents the speed at which electromagnetic waves, such as light, travel through a vacuum. This speed is fundamental to many areas of physics.

• The speed of light, \( c \), is approximately \( 3.00 \times 10^8 \) meters per second (m/s).
• It is often used in equations involving electromagnetic waves, such as \( E = \frac{hc}{\lambda} \), the formula for photon energy.

Knowing the speed of light allows us to understand and calculate how light and other electromagnetic waves propagate through space, helping solve problems ranging from simple wave calculations to complex theories in physics.

Energy of Photon Calculation

The energy of a photon can be calculated using the formula \( E = \frac{hc}{\lambda} \).
This formula highlights the relationship between the energy of a photon, Planck's constant, the speed of light, and the wavelength of the photon.

• \( E \) is the energy of a single photon.
• \( h \) stands for Planck's constant, and \( c \) is the speed of light.
• \( \lambda \) represents the wavelength of the photon.

For instance, using a wavelength \( \lambda = 575 \times 10^{-9} \text{ m} \), the energy \( E \) is calculated as:\[ E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{575 \times 10^{-9}} \]The result gives the amount of energy in joules for one photon. In this problem, it is approximately \( 3.45 \times 10^{-19} \text{ J/photon} \).
This equation is essential when dealing with the interaction of light and matter, allowing us to determine how much energy is carried by light of a certain wavelength.