The wavelengths of an electron beam and X-rays are significantly different. The wavelength of an electron beam is determined by its energy, while X-rays have much shorter wavelengths.
Electron beams can exhibit wave-like properties due to the wave-particle duality of electrons. The de Broglie wavelength, which describes the wavelength associated with a moving particle, is given by the equation:
λ = h / p,
where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. The momentum of an electron is given by its mass (m) times its velocity (v). Therefore, the wavelength of an electron beam depends on its velocity and mass.
In practice, electron beams used in electron microscopes or particle accelerators can have velocities close to the speed of light (c), but typically much lower. For electron beams with velocities commonly used in these applications, the wavelengths range from a few picometers to a few nanometers. For example, an electron beam with an energy of 100 keV (kiloelectron volts) has a wavelength of about 0.03 nm (30 picometers).
On the other hand, X-rays have much shorter wavelengths. X-rays are a form of electromagnetic radiation, similar to light but with higher energy and shorter wavelengths. The exact wavelength of X-rays can vary depending on their source and energy level. However, X-ray wavelengths typically range from about 0.01 nm to 10 nm. For example, the commonly used copper Kα X-ray radiation has a wavelength of 0.154 nm (154 picometers).
In summary, the wavelengths of an electron beam and X-rays are not equal. Electron beams have longer wavelengths, typically ranging from picometers to a few nanometers, while X-rays have much shorter wavelengths, ranging from picometers to tens of nanometers.