(a) what voltage must be applied to an X-ray tube to obtain0.0100 fm wavelength X-rays for use in exploring the details of nuclei? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

A waveform signal that is carried in space or down a wire has a wavelength, which is the separation between two identical places (adjacent crests) in the consecutive cycles.

(a)

The wavelength is \(\lambda = 0.0100\,{\rm{fm}} = 0.0100 \times {10^{ - 15}}\,{\rm{m}}\)

Hence, the energy of the photon is

\({E_\lambda } = \frac{{hc}}{\lambda }\)

Here h is the plank constant (6.62 x 10^-34J.s); c is the speed of light (\(3 \times {10^8}\,{\rm{m/s}}\)) and \(\lambda \) is the wavelength (\(0.0100 \times {10^{ - 15}}\,{\rm{m}}\)).

Substitute all the value in the above equation.

\(\begin{array}{c}{E_\lambda } = \frac{{\left( {6.62 \times {{10}^{ - 34}}\,{\rm{J}}{\rm{.s}}} \right)\left( {3 \times {{10}^8}\,{\rm{m/s}}} \right)}}{{\left( {0.0100 \times {{10}^{ - 15}}\,{\rm{m}}} \right)}}\\ = 1.98 \times {10^{ - 8}}\,{\rm{J}}\\ = \frac{{1.98 \times {{10}^{ - 8}}\,{\rm{J}}}}{{1.6 \times {{10}^{ - 19}}\,{\rm{J/eV}}}}\\ = 1.24 \times {10^{11}}\,{\rm{eV}}\end{array}\)

Hence, the minimum voltage required to accelerate the electron is 1.24 x 10¹¹eV.

(b)

The unreasonable thing about the answer is the requirement of extremely high voltage.

(c)

We are considering electron to generate such a small wavelength. Since the electrons are very light, the momentum will be very small and, hence, making wavelength such small will require huge energy.