To solve the problem step by step, we will address each part of the question sequentially.
### Given Data:
- Speed of the electron, \( v = 2.2 \times 10^6 \, \text{m/s} \)
- Radius of the orbit, \( r = 5.3 \times 10^{-11} \, \text{m} \)
- Charge of the electron, \( e = 1.6 \times 10^{-19} \, \text{C} \)
### (a) Find the orbital period of the electron.
The orbital period \( T \) can be calculated using the formula:
\[
T = \frac{2 \pi r}{v}
\]
Substituting the values:
\[
T = \frac{2 \pi (5.3 \times 10^{-11})}{2.2 \times 10^6}
\]
Calculating the numerator:
\[
2 \pi (5.3 \times 10^{-11}) \approx 3.34 \times 10^{-10} \, \text{m}
\]
Now, dividing by the speed:
\[
T = \frac{3.34 \times 10^{-10}}{2.2 \times 10^6} \approx 1.52 \times 10^{-16} \, \text{s}
\]
Thus, the orbital period of the electron is:
\[
\boxed{1.52 \times 10^{-16} \, \text{s}}
\]
### (b) Find the current \( I \) due to the orbiting electron.
The current \( I \) can be calculated using the formula:
\[
I = \frac{e}{T}
\]
Substituting the values:
\[
I = \frac{1.6 \times 10^{-19}}{1.52 \times 10^{-16}}
\]
Calculating the current:
\[
I \approx 1.05 \times 10^{-3} \, \text{A}
\]
Thus, the current is:
\[
\boxed{1.05 \times 10^{-3} \, \text{A}}
\]
### (c) Find the magnetic moment \( \mu \) of the atom due to the motion of the electron.
The magnetic moment \( \mu \) can be calculated using the formula:
\[
\mu = I \cdot A
\]
where \( A \) is the area of the circular orbit, given by:
\[
A = \pi r^2
\]
Calculating the area:
\[
A = \pi (5.3 \times 10^{-11})^2 \approx 8.84 \times 10^{-21} \, \text{m}^2
\]
Now substituting the values to find the magnetic moment:
\[
\mu = (1.05 \times 10^{-3}) \cdot (8.84 \times 10^{-21})
\]
Calculating the magnetic moment:
\[
\mu \approx 9.29 \times 10^{-24} \, \text{A m}^2
\]
Thus, the magnetic moment of the atom is:
\[
\boxed{9.29 \times 10^{-24} \, \text{A m}^2}
\]
