Video Transcript
A photon has a momentum of 1.11
times 10 to the negative 33 kilogram meters per second. What is the frequency of the
photon? Use a value of 6.63 times 10 to the
negative 34 joule-seconds for the Planck constant. Give your answer to the nearest
megahertz.
To answer this question, we will
need to calculate the frequency of a photon knowing its momentum and then report
this frequency in units of megahertz.
The momentum of a photon can be
expressed as 𝑃, the photon’s momentum, is equal to ℎ, the Planck constant, times
𝑓, the photon’s frequency, divided by 𝑐, the speed of light in vacuum. Since we know the momentum and are
trying to find the frequency, let’s rearrange this formula by multiplying both sides
by 𝑐 divided by ℎ. On the left-hand side, we just have
momentum times speed of light divided by the Planck constant. On the right-hand side, 𝑐 divided
by 𝑐 is one and ℎ divided by ℎ is also one, so we are just left with frequency.
We are given a value for the
photon’s momentum and also a value for the Planck constant. So the last thing that we need is a
value for the speed of light in vacuum. For the purposes of this question,
it is sufficient to use a value for the speed of light of 3.00 times 10 to the
eighth meters per second. This value is always a good
starting point for the speed of light. And if we go through the
calculation and discover that our result is not sufficiently accurate, all we need
to do is perform the same calculation again but with a more accurate value for the
speed of light.
Nevertheless, for this particular
question, we already know that two decimal places is sufficient because both of the
values we’re given in the question are only accurate to two decimal places, which
means two decimal places of accuracy is all we need.
Substituting values, we have 1.11
times 10 to the negative 33 kilogram meters per second times 3.00 times 10 to the
eighth meters per second divided by 6.63 times 10 to the negative 34
joule-seconds. Kilogram meters per second times
meters per second is kilograms meter squared per second squared, which is exactly
one joule. So the units in the numerator are
joules, and the units in the denominator are joules times seconds. Joules divided by joule-seconds is
just inverse seconds, which are hertz. So the overall units of this
quantity is hertz, which tells us we are on the right track because hertz is a unit
for frequency.
Evaluating the numerical portion of
this calculation gives us 502.262 and several more decimal places times 10 to the
sixth with units of hertz. Now remember we want our answer in
units of megahertz. So we recall that one megahertz is
equal to 10 to the sixth hertz. Looking back at our answer, we see
that it is already expressed as a mantissa times 10 to the sixth hertz. So we can just replace 10 to the
sixth hertz with megahertz to get our answer in appropriate units.
Finally, we want our answer to the
nearest megahertz, which means rounding 502.262 et cetera to the nearest
integer. Well, the nearest integer to
502.262 et cetera is just 502. So the frequency of our photon is
502 megahertz. This frequency happens to be part
of the ultra high frequency radio wave portion of the electromagnetic spectrum.
