Show that the cutoff wavelength (in picometers) in the...

Show that the cutoff wavelength (in picometers) in the continuos x-ray spectrum from any target is given by $\lambda_{\min }=1240 / V$ , where $V$ is the potential difference (in kilovolts) through which the electrons are accelerated before they strike the target.

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Key Concepts

Electron Kinetic Energy

When electrons are accelerated by a potential difference, they gain kinetic energy equal to the product of the charge of an electron and the applied voltage. This energy determines the maximum amount available for conversion into x-ray photon energy upon collision with a target.

Photon Energy-Wavelength Relationship

A photon’s energy is inversely related to its wavelength, as expressed by the relation E = hc/?, where h is Planck’s constant and c is the speed of light. This relationship shows that the highest photon energy (or the shortest wavelength) is obtained when the entire kinetic energy of an electron is converted into a single photon.

Unit Conversion Using Fundamental Constants

The constant value of 1240 comes from the product of Planck’s constant and the speed of light, expressed in suitable units (typically electron-volt nanometers). This conversion factor is critical for relating the electron’s kinetic energy, given in kiloelectronvolts, to the cutoff wavelength of the produced x-rays in picometers.

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