The average current due to an electron orbiting the proton in the `n^(th)` Bohr orbit of the hydrogen atom is

Options

A. `propn`
B. `propn^(3)`
C. `propn^(-3)`
D. none of these

Answer

To find the average current due to an electron orbiting the proton in the \( n^{th} \) Bohr orbit of the hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship between radius and quantum number
The radius of the \( n^{th} \) Bohr orbit is given by the formula:
\[ R_n \propto \frac{n^2}{Z} \]
For hydrogen, \( Z = 1 \), so we can simplify this to:
\[ R_n \propto n^2 \]

### Step 2: Determine the velocity of the electron in the orbit
The velocity \( V \) of the electron in the \( n^{th} \) orbit is given by:
\[ V_n \propto \frac{Z}{n} \]
Again, for hydrogen, this simplifies to:
\[ V_n \propto \frac{1}{n} \]

### Step 3: Calculate the angular frequency \( \omega \)
The angular frequency \( \omega \) is defined as:
\[ \omega = \frac{V}{R} \]
Substituting the expressions for \( V \) and \( R \):
\[ \omega \propto \frac{Z/n}{n^2/Z} = \frac{Z^2}{n^3} \]
For hydrogen, this simplifies to:
\[ \omega \propto \frac{1}{n^3} \]

### Step 4: Relate current to charge and time
The average current \( I \) due to the orbiting electron can be expressed as:
\[ I = \frac{q}{T} \]
Where \( q \) is the charge of the electron (denoted as \( e \)) and \( T \) is the time period of one complete revolution. The time period \( T \) can be expressed in terms of angular frequency:
\[ T = \frac{2\pi}{\omega} \]
Substituting this into the current formula gives:
\[ I = e \cdot \omega \cdot \frac{1}{2\pi} \]

### Step 5: Substitute for \( \omega \)
Substituting the expression for \( \omega \):
\[ I \propto e \cdot \frac{Z^2}{n^3} \cdot \frac{1}{2\pi} \]
For hydrogen, this simplifies to:
\[ I \propto \frac{1}{n^3} \]

### Conclusion
Thus, the average current due to an electron orbiting the proton in the \( n^{th} \) Bohr orbit of hydrogen is:
\[ I \propto n^{-3} \]

### Final Answer
The average current is inversely proportional to the cube of the principal quantum number \( n \):
\[ I \propto n^{-3} \]

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