Video Transcript
The table shows a set of types of
electromagnetic waves and the orders of magnitude of their wavelengths. Now this question has two parts to
it. The first part asks as using the
values from the table, how many wavelengths of ultraviolet radiation would have the
same total length as one wavelength of infrared radiation? Answer in standard form.
So this part of the question is
asking us to work out how many wavelengths of ultraviolet radiation would have the
same total length as one wavelength of infrared radiation and we need to give our
answer in standard form. So we’re comparing the wavelengths
of infrared radiation — that’s 10 to the negative five meters — and ultraviolet
radiation, which is 10 to the negative eight meters. What the question is asking us is
if this is one wavelength of infrared radiation, then how many wavelengths of
ultraviolet radiation is that equivalent to?
Now, we know that this distance, a
single wavelength of infrared radiation, is 10 to the negative five meters. And we know that a single
wavelength of ultraviolet radiation is 10 to the negative eight meters. So how many lots of 10 to the
negative eight meters fit into one lot of 10 to the negative five meters? We can work this out by saying that
10 to the negative five — that’s one wavelength of infrared radiation — is equal to
𝑛 times 10 to the negative eight, where 10 to the negative eight is one wavelength
of ultraviolet radiation and 𝑛 is the total number of wavelengths that fit inside
one wavelength of infrared radiation.
So at this point, we’ve got an
equation that links together the infrared wavelength and ultraviolet wavelength and
the number of wavelengths of ultraviolet that fit into one wavelength of
infrared. So we can rearrange to find out the
value of 𝑛. We can do this by dividing both
sides of the equation by 10 to the power of negative eight. That leaves us with the value of 𝑛
on the right-hand side. And 10 to the negative five divided
by 10 to the power of negative eight is the same as 10 to the power of three. That already is in standard
form. And so that’s our answer to the
first part of the question.
In other words, 10 to the power of
three wavelengths of ultraviolet radiation have the same total length as one
wavelength of infrared radiation. So there are one, two, three, four,
five, and so on and so forth. And there are 10 to the power of
three of these in one wavelength of infrared. So clearly, our diagram is not
quite accurate because we haven’t quite drawn 10 to the power of three or 1000
little blue wavelengths. However, we’ve been able to work
out the answer. So that’s a good thing.
Let’s therefore move on to the next
part of the question. This part of the question asks us
“using the values from the table, how many wavelengths of the longest wavelength
gamma radiation would have the same total length as one wavelength of X-ray
radiation? Answer in standard form.”
So here, we’re doing a very similar
thing to what we’ve already done, except this time we’re comparing the wavelengths
of the longest wavelength gamma radiation and one wavelength of X-ray radiation. Once again, we need to give our
answer in standard form. So here, we’re comparing X-ray
radiation which is 10 to the negative 10 meters and gamma rays which have
wavelengths of less than 10 to the negative 15 meters. So if gamma rays have wavelengths
less than 10 to the negative 15 meters, then the longest possible gamma ray must
have a wavelength of 10 to the negative 15 meters cause anything that has a
wavelength larger than this value is not a gamma ray, whereas anything less than
this value is a gamma ray.
So once again, we’re comparing one
wavelength of X-ray radiation to lots of wavelengths of gamma radiation. And we know that the wavelengths in
question are 10 to the negative 10 meters for X-rays and 10 to the negative 15
meters for gamma rays. So yet again, we can say that 10 to
the negative 10 is equal to — and this time we use the letter 𝑚 to represent the
number of wavelengths of gamma radiation in one wavelength of X-ray radiation
because we used the letter 𝑛 earlier. So 10 to the negative 10 is equal
to 𝑚 times 10 to the negative 15.
So once again we rearrange and we
find that the left-hand side of the equation becomes 10 to the negative 10 divided
by 10 to the negative 15. And thus, we find a value of 𝑚 of
10 to the power of five and once again this is in standard form. So our final answer is that 10 to
the power of five wavelength of the longest wavelength gamma radiation would have
the same total length as one wavelength of X-ray radiation.
