The figure shows the relation
between wavelength and intensity of the X-ray spectrum produced by a Coolidge
tube. Which wavelengths listed below
depend on the potential difference between the filament and the target? a) 𝜆 one,
𝜆 two; b) 𝜆 two, 𝜆 three; c) 𝜆 one, 𝜆 four; d) 𝜆 one, 𝜆 three.
Looking at this figure, we’re told
that it represents the X-ray spectrum of radiation produced by a Coolidge tube. The figure shows the radiation
intensity of the X-rays produced compared to their wavelengths. On the horizontal axis, we see four
wavelengths in particular that are identified. And the question asks which of
those wavelengths depend on the potential difference between the filament and the
target in the Coolidge tube.
To get started, let’s recall the
construction of a Coolidge tube. A Coolidge tube consists of an
evacuated glass chamber that contains a cathode as well as an anode. The cathode is a source of
electrons which are accelerated towards the anode thanks to a potential difference
set up between the two that we can call Δ𝑉.
When these accelerated electrons
smash into the target anode, X-rays are produced from the collision. It’s these X-rays that create the
data which is plotted in our figure. Each of these rays has a certain
wavelength as well as a certain radiation intensity. Our question asks which of the
wavelengths listed on our horizontal axis depend on the potential difference Δ𝑉
that set up between the anode and the cathode that is the filament in the target of
our Coolidge tube.
We can get started answering this
question by thinking in terms of energy. Considering the horizontal axis,
which of these four wavelengths corresponds to the highest energy X-ray? We know that the energy of a photon
such as an X-ray is equal to Planck’s constant multiplied by the speed of the photon
divided by its wavelength. This tells us that the smaller the
wavelength, the greater the energy of our photon.
Looking over the wavelength on our
figure, we see that 𝜆 one corresponds to the smallest wavelength and therefore to
the highest energy X-ray. On top of that, the radiation
intensity curve seems to be dropping off rapidly at 𝜆 one. That is, it appears that the X-rays
that have a wavelength 𝜆 one are the highest energy X-rays emitted by this
This can lead us to the question
“Where did these X-rays get their energy?” And we know the answer is they got
their energy from the electrons which were crashing into the target, the anode. If we think about those electrons
as they’re at the tip of the cathode ready to be accelerated across the gap, we knew
that they have some amount of potential energy. That’s because the electrons have a
charge; we’ll call it 𝑞. And they’re also exposed to a
potential difference; we’ve called it Δ𝑉.
The product of these two terms is
equal to the electrical potential energy of the electrons. And we know that the energy doesn’t
remain as potential. We know that as the electrons are
put in motion, it’s converted to kinetic energy which itself then becomes the source
of the energy which produces X-rays.
The highest energy X-rays produced
— those with a wavelength of 𝜆 one — have their energy thanks to the accelerating
electrical potential of 𝑞Δ𝑉. The particular value of 𝜆 one
corresponding to that maximum energy, therefore, depends on the potential difference
Δ𝑉. We can say then that 𝜆 one, the
shortest wave wavelength listed, does depend on this potential difference. As we consider our answer choices,
that eliminates answer option b which doesn’t include 𝜆 one.
Let’s continue on looking at the
other wavelengths written on this axis. If we look at 𝜆 two and 𝜆 three,
we see these wavelengths correspond to peaks in this radiation intensity curve. Those peaks likely have to do with
the particular spectral characteristics of the target material we’re using in our
Coolidge tube. That material which may be tungsten
or copper or some other element has a particular spectral profile which is revealed
through the X-ray radiation.
But we know that that spectrum
which leads to the creation of these figures in the radiation intensity curve
doesn’t have to do with Δ𝑉, the potential difference. Rather, it has to do with the
Lastly then, let’s consider 𝜆
four, the longest and therefore the lowest energy wavelength marked out here. If we look at the point on our
curve corresponding to 𝜆 four, we see that this point doesn’t involve an energy
maximum that was 𝜆 one and it doesn’t involve any particular feature of the
curve. In fact, it doesn’t even represent
the energy minimum because we see our Curve continues on to higher wavelength and
But there is something special
about this wavelength, 𝜆 four. If we go straight up from this
wavelength until we reach our curve, then straight horizontally until we arrive at
the vertical axis, we arrive at a point that we can give a label. It’s the radiation intensity
corresponding to 𝜆 four. It’s this particular value which
identifies 𝜆 four as a special wavelength.
Going back briefly to our sketch of
the Coolidge tube, if the potential difference Δ𝑉 were to increase or to decrease,
that would create a change in the shape of our radiation intensity curve. And under those conditions, this
particular wavelength 𝜆 four would probably not correspond to this particular
radiation intensity, 𝑅 sub 𝜆 four.
The fact that in our scenario these
two values do correspond to one another has to do with the potential difference
between the filament and the target in our tube. If that potential difference were
higher or lower than it is, this correspondence likely would not exist. In its own way then, 𝜆 four is
also dependent on the potential difference Δ𝑉.
And knowing that, we now know how
to make our answer selection. We choose option c which includes
both 𝜆 one and 𝜆 four as the X-ray wavelengths which depend on the potential
difference Δ𝑉 between the filament and the target of our tube.