Question

In the Bohr’s model of the hydrogen atom, the electron is pictured to rotate in a circular orbit of radius $5 \times {10^{ - 11}}m$ , at a speed $2.2 \times {10^6}$ . What is the current associated with electron motion in $mA$ ?

Answer 1

Hint: First use the equation $s = \dfrac{{2\pi r}}{t}$ to calculate the value of the time period of revolution $t$ of the electron. Then substitute the value of $t$ in the equation \[Current = \dfrac{{Charge{\text{ }}of{\text{ }}an{\text{ }}electron}}{{Time{\text{ }}period{\text{ }}of{\text{ }}revolution{\text{ }}of{\text{ }}an{\text{ }}electron}}\] to calculate the value of current and remember to convert it to $mA$ $\left( {{{10}^{ - 6}}A = 1mA} \right)$ .

Complete step by step answer:
Bohr’s model of a hydrogen atom depicts the structure of an atom in reference to a hydrogen atom. According to Bohr’s model of a hydrogen atom in the center of an atom lies a nucleus that contains all the nucleons (the protons and the neutrons). The electrons revolve around the nucleus in fixed circular stationary objects, in these stationary orbits, the electrons have a fixed speed and hence angular momentum.
First, we have to calculate the time period of revolution of the electron around the nucleus.
The equation for the time period of an electron is
$s = \dfrac{{2\pi r}}{t}$
$s = $ Speed of the electron
$r = $ The radius of the orbit in which the electron revolves
$t = $ The time period of revolution of an electron
So, for this problem the equation becomes
$2.2 \times {10^6} = \dfrac{{2 \times \pi \times 5 \times {{10}^{ - 11}}}}{t}$
$t = \dfrac{{10}}{7} \times {10^{ - 16}}\sec $
We also know that the magnitude of the charge on an electron is \[1.6 \times {10^{ - 19}}C\] and it is given that it requires $\dfrac{{10}}{7} \times {10^{ - 16}}\sec $ to complete a full revolution around the nucleus.
So, the current associated with the motion of an electron is given by the equation
\[Current = \dfrac{{Charge{\text{ }}of{\text{ }}an{\text{ }}electron}}{{Time{\text{ }}period{\text{ }}of{\text{ }}revolution{\text{ }}of{\text{ }}an{\text{ }}electron}}\]
\[Current = \dfrac{{1.6 \times {{10}^{ - 19}}}}{{\dfrac{{10}}{7} \times {{10}^{ - 16}}}}\]
\[Current = 11.2 \times {10^{ - 5}}A = 1.12mA\]
So, the current associated with the electron is \[1.12mA\] .

Note: Bohr’s model of a hydrogen atom is important but had a lot of shortcomings too. It could not explain the presence of multiple spectral lines of the hydrogen atom, it could explain the dual nature of matter, and it could not explain the uncertainty principle and the quantization of energy.