Question

The wavelength of an X-ray beam is 10 Å. The mass of a fictitious particle having the same energy as that of the X-ray photons is (x/3) h kg. The value of x is _______.

Answer 1

Hint: According to plank’s law, every radiation from a source of light is in form of small packets called as photons. Each photon has an energy, \[E = h{\mathbf{v}}\]where v is the frequency of radiation, h is the plank’s constant having value $6.63{(10)^{ - 34}}$Js. The law signifies that energy is directly proportional to frequency of radiation. On increasing the frequency of radiation, energy also increase ad vice versa. Also, the energy is given by newton’s equation relating mass and energy as $E = m{c^2}$.thus, energy is directly proportional to mass.

Complete answer:
\[E = h{\mathbf{v}}\]
\[E = h\frac{c}{{\mathbf{\lambda }}}\]
Given the value of wavelength,
\[\lambda = 10{\overset{\circ}{A}} = 10({10^{^{ - 10)}}}\]m
\[\lambda = {10^{^{^{ - 9}}}}\]

Thus,
\[{\mathbf{E}} = {\mathbf{h}}\frac{c}{{{{10}^{ - 9}}}}\] ---------(1)
Also, $E = m{c^2}$------(2)

Equating the above two equation
\[E = h\frac{c}{{{{10}^{ - 9}}}}\]$ = m{c^2}$
\[mc = \frac{h}{{{{10}^{ - 9}}}}\]
\[m = \frac{h}{{c{{10}^{ - 9}}}}\]
Using Speed of light is \[3{(10)^8}m{s^{ - 1}}\]
\[m = \frac{h}{{3{{(10)}^8}{{10}^{ - 9}}}}\]

Given value of mass is
\[\left( {x/3} \right){\text{ }}h{\text{ }}kg\]

Putting it in the above equation
\[\left( {x/3} \right){\text{ }}h = \frac{h}{{3{{(10)}^8}{{10}^{ - 9}}}}\]
 \[\left( {x/3} \right){\text{ }} = \frac{1}{{3{{(10)}^8}{{10}^{ - 9}}}}\]{Cancelling h from both sides}
$x = 10$

Thus, the value of x is $10$kg.

Note: The Einstein law signifies that mass and energy are interconvert able and as energy is always conserved(law of conservation of energy) so is the mass. Even when the body is at rest , it has a potential energy available to convert to other form equal to mass into velocity square. This law signifies the particle nature of radiation as photons are tiny particles. The reciprocal of wavelength is called wave number hence the planks law can be also written as \[{\mathbf{E}} = \frac{h}{\nu }\].ν stands for wave number.