An electron in a hydrogen in a hydrogen atom makes a transition from first excited state to ground state. The equivelent current due to circulating electron

Options

A. decreases 16 times
B. decreases 8 times
C. increases 8 times
D. increases 32 times

Explanation

To solve the problem of finding the equivalent current due to a circulating electron in a hydrogen atom as it transitions from the first excited state (n=2) to the ground state (n=1), we can follow these steps:

### Step-by-Step Solution

1. **Understanding Current Due to Circulating Charge**:
The current \( I \) due to a circulating charge can be defined as:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge and \( T \) is the time period for one complete revolution.

2. **Finding the Time Period**:
The time period \( T \) for one revolution can be expressed as:
\[ T = \frac{2\pi R}{V} \]
where \( R \) is the radius of the orbit and \( V \) is the velocity of the electron.

3. **Relating Radius and Velocity to Quantum Number**:
In a hydrogen atom, the radius \( R \) of the electron's orbit is proportional to the square of the principal quantum number \( n \):
\[ R \propto n^2 \]
The velocity \( V \) of the electron is inversely proportional to the principal quantum number \( n \):
\[ V \propto \frac{1}{n} \]

4. **Substituting into the Current Formula**:
Substituting \( R \) and \( V \) into the expression for \( T \):
\[ T \propto \frac{2\pi n^2}{\frac{1}{n}} = 2\pi n^3 \]
Therefore, the current can be expressed as:
\[ I \propto \frac{Q}{T} \propto \frac{Q}{n^3} \]

5. **Current Ratio for Different States**:
Let \( I_1 \) be the current in the first excited state (n=2) and \( I_0 \) be the current in the ground state (n=1):
\[ \frac{I_1}{I_0} = \frac{n_0^3}{n_1^3} = \frac{1^3}{2^3} = \frac{1}{8} \]
Thus, we find that:
\[ I_1 = 8 I_0 \]

6. **Conclusion**:
Therefore, when the electron transitions from the first excited state (n=2) to the ground state (n=1), the equivalent current increases by a factor of 8. The final answer is:
\[ I_{\text{final}} = 8 I_{\text{initial}} \]

### Final Answer
The equivalent current due to the circulating electron when it transitions from the first excited state to the ground state is 8 times the initial current.