If the power of light is 10 kW and the wavelength is 300 nm, find the number of photons per second.

Understand the Problem

The question is asking how to calculate the number of photons emitted per second by a light source with a power of 10 kW and a wavelength of 300 nm. This involves using the relationship between energy, power, and the wavelength of light to find the total number of photons.

Answer

The number of photons emitted per second is approximately \(2.01 \times 10^{20}\).
Answer for screen readers

The number of photons emitted per second is approximately (2.01 \times 10^{20}).

Steps to Solve

  1. Determine the energy of a single photon

The energy $E$ of a photon can be calculated using the formula:

$$ E = \frac{hc}{\lambda} $$

where:

  • $h$ is Planck's constant ($6.626 \times 10^{-34} , \text{Js}$),
  • $c$ is the speed of light in a vacuum ($3.00 \times 10^8 , \text{m/s}$), and
  • $\lambda$ is the wavelength ($300 , \text{nm}$, which is $300 \times 10^{-9} , \text{m}$).
  1. Calculate the energy per photon

Substituting in the values:

$$ E = \frac{(6.626 \times 10^{-34} , \text{Js})(3.00 \times 10^8 , \text{m/s})}{300 \times 10^{-9} , \text{m}} $$

Calculate this value to find the energy of a single photon.

  1. Calculate the total energy emitted per second

Using the power of the light source ($P = 10 , \text{kW} = 10,000 , \text{W}$):

$$ P = \frac{\text{Total Energy}}{\text{Time}} $$

Since time is 1 second, the total energy emitted per second is equal to the power:

$$ \text{Total Energy} = 10,000 , \text{J} $$

  1. Find the number of photons emitted per second

The number of photons $N$ emitted per second can be found using:

$$ N = \frac{\text{Total Energy}}{E} $$

Now substitute in the total energy from the previous step and the energy per photon from step 2.

  1. Calculate the final value

Perform the calculation:

$$ N = \frac{10,000 , \text{J}}{E} $$

This will give you the number of photons emitted per second.

The number of photons emitted per second is approximately (2.01 \times 10^{20}).

More Information

This calculation illustrates how energy and wavelength relate to the quantity of light emitted from a source. The wavelength is critical in determining the energy of each photon and, in turn, how many are produced based on the power output of the light source.

Tips

  • Confusing power (W) and energy (J): Remember that power is the rate of energy emitted over time.
  • Incorrectly converting units, especially wavelength from nm to m; always ensure unit consistency.

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