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 our figure, we see that it’s a plot of the spectral radiant intensity of
X-rays emitted by a Coolidge tube. There are four particular wavelength values marked out on this figure. And we want to choose which of these values are determined by the potential
difference that exists in the Coolidge tube between the filament and the target.
We’ll recall that a Coolidge tube is used to produce X-rays and that it consists of a
glass chamber with a cathode and anode inside. Thanks to a potential difference, Δ𝑉, that’s set up between the cathode and the
anode, electrons are drawn off of the cathode’s surface and accelerated toward the
anode, the target. When these accelerated electrons smash into the target material, they emit
X-rays. And it’s the spectrum of those X-rays that we’re seeing in our figure.
The maximum energy that these X-rays can have depends on the kinetic energy of the
electrons right before they run into the target material. Written out, we can say that the maximum energy of the X-rays is equal to the final
kinetic energy of the electrons and that this final kinetic energy is equal to the
potential difference, Δ𝑉, multiplied by the charge of an electron, lowercase 𝑞. This shows us how the energy of the X-rays depends on the potential difference,
And if we further recall that the energy of a photon, which an X-ray is, is equal to
Planck’s constant times the speed of the photon, the speed of light, divided by its
wavelength, then we can say that the maximum X-ray energy also is equal to Planck’s
constant times the speed of light, 𝐶, divided by the minimum wavelength that these
photons can have.
In this way then, the potential difference, Δ𝑉, helps to determine what the minimum
wavelength of the emitted X-rays is. Looking at our figure, we see that this minimum wavelength is marked as 𝜆 one. That’s the shortest wavelength and therefore the highest-energy X-ray emitted by this
When we consider this overall process of electrons being accelerated and then
crashing into a target material in order to emit a particular type of radiation, we
know that X-rays, being a fairly high-energy variety of photon, will require a
certain electron energy minimum in order to be generated in the first place. That minimum electron energy required to generate X-rays is largely determined by the
potential difference between the filament and the target.
If that potential difference is too low, the electrons won’t accelerate enough and
therefore when they collide with a target won’t produce any X-rays. So it’s not only the highest-energy X-rays that are influenced by that potential
difference, but also the lowest-energy X-rays produced, the ones with the longest
At this, the bottom end of our energy spectrum, we’re relying on the potential
difference to meet a minimum value required for the generation of X-rays. 𝜆 one and 𝜆 four are therefore the two wavelengths which depend on this potential
difference. And that’s answer option c.