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Kinetic energy is the energy an object has due to its motion. In the context of x-ray production, both protons and electrons gain kinetic energy when they are accelerated by a potential difference. This energy is crucial because it is converted into electromagnetic radiation, such as x-rays, when the particles strike a metal target.
For a proton or electron initially at rest and accelerated through a potential difference of 4,000 volts (4 kV), the kinetic energy acquired can be calculated using the formula:

• \( E = e \times V \)
• Where \( e \) is the charge of the particle (about \( 1.6 \times 10^{-19} \text{C} \)), and \( V \) is the potential difference.

This relationship allows us to calculate that both the proton and electron gain the same amount of kinetic energy, allowing their different masses to impact later processes differently.

When a proton or electron hits a metal target, the particle's kinetic energy is transformed into a photon. The wavelength of this photon is inversely related to its energy. This is described by the equation:

• \( E_{photon} = \frac{hc}{\lambda} \)
• Where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{Js} \)) and \( c \) is the speed of light (\( 3.00 \times 10^8 \text{m/s} \)).

By rearranging the equation, we find the wavelength \( \lambda \) of the photon:

• \( \lambda = \frac{hc}{E} \)

Applying the same kinetic energy value for both protons and electrons yields the same minimum wavelength. However, the efficiency and practicality of using either for x-ray production differ.

Electrons are negatively charged particles and are relatively easy to accelerate. With their much smaller mass compared to protons, electrons reach higher velocities under the same potential difference. This makes them highly effective in x-ray tubes.
Due to being lighter, electrons require less energy to reach sufficient speeds to produce x-rays. This means:

• The machinery used is smaller and less costly.
• The electrons can be controlled more efficiently, improving x-ray production.
• Both Ease of acceleration and smaller size make electrons the preferred choice for x-ray generation.

Overall, these advantages make electron-accelerated machines more common in medical and scientific settings.

Protons, in contrast to electrons, are heavier and positively charged. When accelerated by the same potential difference, protons gain the same amount of kinetic energy as electrons. However, because they are heavier, they do not achieve the high velocities that electrons do.
This difference manifests in several practical challenges:

• Proton acceleration requires larger and more advanced equipment, leading to higher costs.
• Protons are harder to handle and control efficiently.
• The increased mass means it is more difficult to produce focused and efficient x-ray beams.

These limitations are why protons are not typically used for x-ray production, despite having comparable kinetic energy to electrons.