In an X-ray tube operated at a fixed accelerating potential, replacing the target anode wi… — JAMB UTME

In an X-ray tube operated at a fixed accelerating potential, replacing the target anode with a metal of higher atomic number decreases the cut-off (minimum) wavelength $\lambda_{\text{min}}$ of the continuous X-ray spectrum.

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False

The statement is false because, by the Duane-Hunt law (

eV = \frac{hc}{\lambda_{\text{min}}}

), the minimum cut-off wavelength depends strictly on the accelerating potential

V

applied across the tube and fundamental physical constants. It is completely independent of the target material's atomic number

Z

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Identify the relationship governing the minimum (cut-off) wavelength of continuous X-rays.

By the Duane-Hunt law, the maximum energy of an emitted X-ray photon equals the total kinetic energy of an accelerating electron:

E_{\text{max}} = eV = \frac{hc}{\lambda_{\text{min}}}

The continuous X-ray spectrum is produced via Bremsstrahlung (braking radiation) when high-speed electrons are decelerated by the electric fields of target nuclei.

Express the minimum wavelength

\lambda_{\text{min}}

in terms of operational variables.

Rearranging the equation yields

\lambda_{\text{min}} = \frac{hc}{eV}

, where

h

is Planck's constant,

c

is the speed of light,

e

is the elementary charge, and

V

is the tube potential.

This formula establishes that

\lambda_{\text{min}}

depends exclusively on the accelerating voltage

V

and physical constants.

Analyze the effect of altering the target material's atomic number

Z

at constant voltage

V

Changing the atomic number

Z

shifts characteristic spectral line wavelengths (via Moseley's law) and increases overall Bremsstrahlung intensity, but leaves

\lambda_{\text{min}}

entirely unchanged.

Since

Z

does not enter the Duane-Hunt expression for

\lambda_{\text{min}}

, varying

Z

has zero effect on the shortest emitted wavelength.

Anahtar Kavram

Duane-Hunt Law and Cut-off Wavelength Independence from Target Material

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