Question

The shortest wavelength produced in an X ray tube operating at \[0.5\times {{10}^{6}}\]V is?
A- depending on the target element
B- about \[2.5\times {{10}^{-12}}\]m
C- double of the shortest wavelength produced at 1 million volts
D- dependent only on the target material

Answer 1

Hint: Wavelength is defined as the distance between corresponding points of two consecutive waves. It is an important property of a wave and it depends upon the medium in which it is travelling unlike frequency which does not remain constant.

Complete step by step answer:Step by step solution-
The voltage is given, from Planck radiation formula, \[E=\dfrac{hc}{\lambda }\], where h is the Planck constant and c, is the speed of light. Also, in terms of kinetic energy, \[E=Ve\], where V is the accelerating potential and e is the charge of the accelerating charged particles. X-rays are produced when fast moving electrons are suddenly brought to rest.
So, \[E=\dfrac{hc}{\lambda }\] and \[E=Ve\].
Thus, \[\lambda =\dfrac{hc}{Ve}\]------(1)
This shows the dependency of the wavelength produced on the parameters. If V is doubled, wavelength is halved. Putting the values in eq (1),
\[\lambda =\dfrac{hc}{Ve}=\dfrac{6.6\times {{10}^{-34}}\times 3\times {{10}^{8}}}{0.5\times {{10}^{-6}}\times 1.6\times {{10}^{-19}}}\]
\[\lambda =2.5\times {{10}^{-12}}m\]
So, the correct options are (b) and (C)

Additional information-X-rays are used in diagnosis in drawing the picture of a human's internal organs as they get deflected by the bones but not by the mass. Velocity of an electromagnetic wave is a property which is dependent on the medium in which it is travelling. So, if the medium changes then the speed of the electromagnetic wave will also change, but in a given medium, they all travel with the same speed.

Note: wavelength is the distance between crest to crest or trough to trough of the wave. The wave changes its shape as it moves from one medium to another and as such the wavelength also changes. Contrary to this frequency remains constant.