Question Video: Calculating the Wavelength of a Photon from Its Momentum Physics • 9th Grade

What is the wavelength of a photon that has a momentum of 5.00 × 10⁻²⁵ kg⋅m/s? Use a value of 6.63 × 10⁻³⁴ J⋅s for the Planck constant. Give your answer to two decimal places.

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Video Transcript

What is the wavelength of a photon that has a momentum of 5.00 times 10 to the negative 25 kilogram meters per second? Use a value of 6.63 times 10 to the negative 34 joule-seconds for the Planck constant. Give your answer to two decimal places.

This question asks us to find the wavelength of a photon given its momentum. We can recall that the momentum of a photon is defined as the Planck constant divided by its wavelength. And we can rearrange this formula multiplying both sides by wavelength and dividing both sides by momentum to find the wavelength of a photon is equal to the Planck constant divided by its momentum. Well, we are given a value for the Planck constant and we are given a value for the momentum. So all that’s left to do is substitute values. 6.63 times 10 to the negative 34 divided by 5.00 times 10 to the negative 25 is 1.326 times 10 to the negative nine.

The units for this quantity are joule-seconds divided by kilograms meters per second. We could work out what these units are equivalent to by rewriting joules in terms of kilograms, meters, and seconds, but there is a much simpler way. We are calculating a wavelength, so the overall units must be appropriate for measuring length. Now, joules can be expressed directly in terms of SI base units, and kilograms, meters, and seconds are themselves SI base units. So the combination of all these units must give another SI base unit, this time the SI base unit for length, which is the meter. So the overall units of this quantity are meters. So our wavelength is 1.326 times 10 to the negative nine meters.

When we round 1.326 to two decimal places, we get 1.33. And just to make our answer a little bit neater, we’ll recall that 10 to the negative nine meters is one nanometer. So our final answer is 1.33 nanometers. This wavelength is much smaller than the wavelength of a typical photon of visible light, which has a wavelength of around 500 nanometers. So this photon must be a high-energy X-ray or gamma ray.

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