Video Transcript
What is the momentum of a photon that has a wavelength of 500 nanometers? Use a value of 6.63 times 10 to the negative 34 joule-seconds for the Planck constant. Give your answer in scientific notation to two decimal places.
We are looking for the momentum of a photon. And we are given a value for its wavelength, specifically 500 nanometers, and also a value for the Planck constant. We recall that the momentum of a photon is defined by the Planck constant divided by the photon’s wavelength. We need to define the momentum of a photon this way because photons have no mass. So the usual definition of mass times velocity doesn’t apply to a photon.
Before we substitute values into this formula, note that the units that we have for the Planck constant are joule-seconds, which can be expressed in terms of base SI units. So it will be helpful to make our calculations easier if we express the wavelength 500 nanometers also in terms of base SI units. If we recall that one nanometer is 10 to the negative nine meters, then the wavelength of the photon is 500 times 10 to the negative nine meters, or 5.00 times 10 to the negative seven meters. We’ve expressed the wavelength this way because it is now in scientific notation to two decimal places, which is the proper format for our final answer.
Substituting values for the Planck constant and the wavelength into our formula for the photon momentum, we have 6.63 times 10 to the negative 34 joule-seconds divided by 5.00 times 10 to the negative seven meters, which is equal to 1.326 times 10 to the negative 27 joule-seconds per meter. Since joules are kilograms meters squared per second squared, we can express the units of this quantity as kilograms meter squared per second squared times seconds per meter. Seconds per second squared is just per second, and meters squared per meter is just meters. So the overall units are kilogram meters per second, which tells us that we are on the right track because kilogram meters per second are units for momentum.
Finally, we round 1.326 to two decimal places, which gives us 1.33. And our final answer to two decimal places in scientific notation is 1.33 times 10 to the negative 27 kilogram meters per second.
