The electron of a hydrogen atom revolves the proton in a circuit nth of radius `r_(0) = (in_(0) n^(2)h^(2))/(pi m e^(2)) ` with a speed `upsilon_(0) =(e^(2))/(2 in_(0) nh)` The current the to circulating charge is proportional to

Options

A. `e^(2)`
B. `e^(3)`
C. `e^(5)`
D. `e^(6)`

Explanation

To solve the problem, we need to find the current due to the circulating charge of the electron in a hydrogen atom. The current \( I \) is given by the formula:

\[ I = \frac{Q}{T} \]

where \( Q \) is the charge and \( T \) is the time period of the electron's revolution.

### Step 1: Identify the charge \( Q \)
The charge \( Q \) of the electron is given by:

\[ Q = e \]

### Step 2: Find the time period \( T \)
The time period \( T \) can be calculated using the formula:

\[ T = \frac{\text{Distance}}{\text{Speed}} \]

The distance the electron travels in one complete revolution is the circumference of the circular orbit, which is:

\[ \text{Distance} = 2 \pi r_0 \]

where \( r_0 \) is the radius of the orbit given by:

\[ r_0 = \frac{\epsilon_0 n^2 h^2}{\pi m e^2} \]

The speed \( v_0 \) of the electron is given by:

\[ v_0 = \frac{e^2}{2 \epsilon_0 n h} \]

Now substituting these into the formula for \( T \):

\[ T = \frac{2 \pi r_0}{v_0} \]

Substituting the expressions for \( r_0 \) and \( v_0 \):

\[ T = \frac{2 \pi \left(\frac{\epsilon_0 n^2 h^2}{\pi m e^2}\right)}{\left(\frac{e^2}{2 \epsilon_0 n h}\right)} \]

### Step 3: Simplify the expression for \( T \)
Simplifying the expression:

\[ T = \frac{2 \pi \epsilon_0 n^2 h^2 \cdot 2 \epsilon_0 n h}{\pi m e^2 \cdot e^2} \]

This simplifies to:

\[ T = \frac{4 \epsilon_0^2 n^3 h^3}{m e^4} \]

### Step 4: Substitute \( T \) back into the current formula
Now substituting \( T \) back into the current formula:

\[ I = \frac{e}{T} = \frac{e}{\frac{4 \epsilon_0^2 n^3 h^3}{m e^4}} = \frac{m e^5}{4 \epsilon_0^2 n^3 h^3} \]

### Step 5: Determine the proportionality
From the expression for \( I \), we can see that the current \( I \) is proportional to:

\[ I \propto e^5 \]

### Conclusion
Thus, the current due to the circulating charge is proportional to \( e^5 \).

### Final Answer
The correct option is \( e^5 \).

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