Question

Find the velocity of an electron having an energy of \[1280\] electron volt. ( \[m = 9.1 \times {10^{ - 31}}{\text{kg}}\] )

Answer 1

Hint:We are given the value of energy and mass of the electron and asked to find the velocity. To calculate the velocity, we need to have an equation of energy in terms of velocity, so recall the energy of a moving particle and put the given values to calculate the value of velocity of the electron.

Complete step by step answer:
Given, energy of the electron, \[E = 1280\,{\text{eV}}\].Mass of the electron, \[m = 9.1 \times {10^{ - 31}}{\text{kg}}\].
The formula for energy of a moving particle is,
\[E = \dfrac{1}{2}m{v^2}\] (i)
where \[m\] is the mass of the particle and \[v\] is the velocity of the particle.
Before using the formula, we will check whether the quantities are in the same units. Here, we observe energy is in the unit of electron volt so, we convert the unit of energy to SI unit since mass is in SI unit. SI unit of energy is joule, so we convert electron volt to joule.
\[1\,eV = 1.6 \times {10^{ - 19}}{\text{J}}\]
\[\therefore 1280\,eV = 1280 \times 1.6 \times {10^{ - 19}}{\text{J}}\]
Now, energy of the electron is \[E = 1280 \times 1.6 \times {10^{ - 19}}{\text{J}}\]
We put the values of \[E\] and \[m\] in equation (i) and we get,
\[1280 \times 1.6 \times {10^{ - 19}} = \dfrac{1}{2}\left( {9.1 \times {{10}^{ - 31}}} \right){v^2}\]
\[ \Rightarrow {v^2} = \dfrac{{2 \times 1280 \times 1.6 \times {{10}^{ - 19}}}}{{9.1 \times {{10}^{ - 31}}}}\]
\[ \Rightarrow {v^2} = 450.10 \times {10^{12}}\]
\[ \Rightarrow v = \sqrt {450.10 \times {{10}^{12}}} \]
\[ \Rightarrow v = 21.21 \times {10^6}\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\]
\[ \therefore v = 2.121 \times {10^7}\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\]

Therefore, velocity of the electron is \[2.121 \times {10^7}\,{\text{m}}{{\text{s}}^{{\text{ - 1}}}}\].

Note:In such types of questions, to find the required quantity always try to recall the formula for the required quantity in terms of the given quantity. Also, before proceeding for calculations always check that the units of the quantities are the same, that is whether all quantities are in SI units or in CGS units. If they are not in the same units then convert them to the same units and then proceed for calculations.