Video: Physics Past Exam • 2017/2018 • Pack 1 • Question 12B

Physics Past Exam • 2017/2018 • Pack 1 • Question 12B

05:55

Video Transcript

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 collision process.

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 target material.

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 lower energy.

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.

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