Question

How fast is an electron moving if it has a wavelength equal to the distance traveled in one second?

Answer 1

Hint: To solve this question, first analyze the question and determine what has been mentioned in the question. Then using the formula that speed is equal to distance per time, the wavelength in terms of velocity can be obtained. Then in the formula of the wavelength in terms of Planck’s constant and momentum has to be used. Then by simplifying the obtained equation, the speed of the electron can be determined.
Formula used: \[{\text{velocity = }}\dfrac{{{\text{Distance}}}}{{{\text{time}}}}\]
The wavelength, \[{{\lambda = }}\dfrac{{\text{h}}}{{\text{p}}}\] where \[h\] is Planck’s constant and \[p\] is momentum.

Complete step by step answer:
According to the question, we have an electron whose wavelength is equal to the distance in one second.
So in mathematical terms, we can write the above-given information as
\[\lambda {\text{ = }}\dfrac{{{\text{Distance}}}}{{1{\text{ s}}}}{{ }},{\text{where}}\lambda {\text{ is the wavelength}}.{\text{ }} \ldots \ldots \ldots \ldots \ldots ..(1)\]
Now we know that the expression distance by time is equal to the velocity. In mathematical expression we get,
\[{\text{velocity = }}\dfrac{{{\text{Distance}}}}{{{\text{time}}}}{\text{ }}\]…………………..$(2)$
So, now compare both the above equations ${\text{(1) and (2)}}$
We get,
$velocity = \lambda $ …………………$(3)$
Now we know the expression \[{{\lambda = }}\dfrac{{\text{h}}}{{\text{p}}}\], where \[h\] is Planck’s constant and \[p\] is momentum.
We know that ${{p = mass \times \text{velocity}}}$
Therefore we get the above expression to be \[{{\lambda = }}\dfrac{{\text{h}}}{{mv}}\]
Where $m$ is the mass and $v$ is the velocity.
Now substitute the equation $(3)$ in the above-mentioned equation. We get,
\[v = \dfrac{h}{{mv}}\]
Now cross multiply the above equation. We get,
\[{v^2} = \dfrac{h}{m}\]
Now, apply the square root on both sides of the question. We get,
\[v = \sqrt {\dfrac{h}{m}} \]
From this, we have obtained that the electron travels with a speed that is equal to the square root of Planck’s constant by mass when it travels with a wavelength equal to the distance traveled in one second.

Note: Planck’s constant ${\text{h = }}\,{\text{6}}{{.626 \times 1}}{{\text{0}}^{{\text{ - 34}}}}{\text{j s}}$ and mass of the electron ${{\text{m}}_e}{\text{ = }}9.1 \times {10^{ - 31}}{\text{Kg}}$ By substituting these values in the expression we have obtained, we can determine the exact value of the speed of the electron traveling with a wavelength equal to the distance in one second. Using Planck’s law one can determine the energy of the photons if their value of frequency is mentioned.