In an industrial non-destructive testing setup, an X-ray tube produces continuous X-radiat… — JAMB UTME

In an industrial non-destructive testing setup, an X-ray tube produces continuous X-radiation with a minimum cut-off wavelength of $3.3 \times 10^{-11}\text{ m}$ . What is the operating accelerating potential difference of the tube in kilovolts ( $\text{kV}$ )? (Take Planck's constant $h = 6.6 \times 10^{-34}\text{ J s}$ , speed of light $c = 3.0 \times 10^8\text{ m s}^{-1}$ , and electron charge $e = 1.6 \times 10^{-19}\text{ C}$ ).

Answer: 37.5 kV

Answer

The operating accelerating potential difference of the tube is 37.5 kV.

According to the Duane-Hunt law, the maximum kinetic energy of electrons hitting the target equal the maximum photon energy produced:

e V = h f_{\text{max}} = \frac{h c}{\lambda_{\text{min}}}

. Rearranging to solve for voltage gives

V = \frac{h c}{e \lambda_{\text{min}}}

. Substituting

h = 6.6 \times 10^{-34}\text{ J s}

c = 3.0 \times 10^8\text{ m s}^{-1}

e = 1.6 \times 10^{-19}\text{ C}

, and

\lambda_{\text{min}} = 3.3 \times 10^{-11}\text{ m}

yields

V = 37,500\text{ V}

, which equals

37.5\text{ kV}

Step-by-Step Solution

Relate the maximum electron kinetic energy to the shortest X-ray photon wavelength using Duane-Hunt law

e V = E_{\text{max}} = \frac{h c}{\lambda_{\text{min}}}

At the Duane-Hunt cutoff limit, the entire kinetic energy of an accelerating electron is converted into a single X-ray photon.

Rearrange the equation to solve for the accelerating voltage

V

V = \frac{h c}{e \lambda_{\text{min}}}

Isolating

V

allows direct calculation from known fundamental constants and the given minimum wavelength.

Substitute the physical constants and calculate the value of

V

in Volts

V = \frac{(6.6 \times 10^{-34}\text{ J s})(3.0 \times 10^8\text{ m s}^{-1})}{(1.6 \times 10^{-19}\text{ C})(3.3 \times 10^{-11}\text{ m})} = 37,500\text{ V}

Carrying out arithmetic with scientific notation powers yields

3.75 \times 10^4\text{ V}

Convert the potential difference from Volts (V) to kilovolts (kV)

37,500\text{ V} = 37.5\text{ kV}

Dividing by 1000 converts potential difference into the requested kilovolt unit.

Key Concept

Duane-Hunt Law and Cut-off Wavelength in X-ray Production

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