The diagram shows a Coolidge tube used for the production of X-rays. The potential difference 𝑉 one is equal to 60 kilovolts, and the potential difference 𝑉 two is equal to 12 volts. What is the maximum energy of X-rays that the tube can produce?
A Coolidge tube can produce X-rays with energies equal to the product of the charge of an electron 𝑞 and the potential difference across the tube 𝑉𝑡. For this problem, we are given two potential differences: 𝑉 one, which is 60 kilovolts, and 𝑉 two, which is only 12 volts. 𝑉 one is the potential difference between the cathode and the anode of the Coolidge tube. The potential difference between the cathode and anode of a Coolidge tube is also called the acceleration potential difference, since it’s responsible for the acceleration of the electrons in the Coolidge tube.
A higher acceleration potential difference means the electrons will have a higher kinetic energy and thus will produce higher-energy X-rays. When we say the potential difference of the tube, we mean the acceleration potential difference. 𝑉 two just powers the cathode coil, heating up the electrons within it and increasing the beam current releasing electrons in a process called thermionic emission. And because of this, the potential difference that’s used to power the cathode coil is called the thermionic potential difference.
The thermionic potential difference, which in this case is only 12 volts, does not contribute to the X-ray energy maximum presented in the equation here. It controls the number of electrons in the beam but not their energy. If 𝑉 two had a potential difference of zero, then it wouldn’t matter how large or how small 𝑉 one was. No electrons would escape, and thus no X-rays would be produced. So we just have to confirm that it is powered, which in this example it is. So for our equation, we just have to look at 𝑉𝑡, which is 𝑉 one, the acceleration potential difference.
This means that the equation for the maximum energy of the X-rays that can be produced in this tube is equal to the product of 𝑞, the charge of an electron, and 𝑉 one, our acceleration potential difference, which is 60 kilovolts. The charge of an electron is a constant and is approximately equal to 1.602 times 10 to the power of negative 19 coulombs. These units work out great for us because the SI unit of energy, the joule, is equal to coulombs times volts.
All we have to do before taking the product here is to get rid of the prefix of kilo- here. One kilovolt is equal to 1000 volts. So to get 𝑉 one in terms of regular volts, we just have to multiply it by 1000. 60 times 1000 is equal to 60000. So plugging in the value of 𝑞 and 𝑉 one means we’re taking the product of 1.602 times 10 to the power of negative 19 coulombs and 60000 volts, which is equal to 9.612 times 10 to the power of negative 15 joules.
This is a technically correct answer, but a power to the negative 15 joules is very, very small. When dealing with the very small energies involved in the Coolidge tube, whether it be the electrons or the X-rays produced, it is common to not use joules but electron volts or eVs. Electron volts are convenient to use because one electron volt is equal to the energy gained by one electron as it is accelerated by one volt of potential difference. This means that the product of 𝑞 and 𝑉 one or 𝑞 times 60000 volts is just 60000 electron volts or 60 kiloelectron volts, which is the maximum energy of X-rays that the Coolidge tube can produce.