In a hydrogen atom of radius `r_n`, an electron is revolving in `n^(th)` orbit with velocity `v_n`. The magnetic field at nucleus of the atom will be

Options

A. `(mu_0ev_n)/(2pir_n^2)`
B. `(mu_0ev_n)/(2pi^2 r_n^2)`
C. `(mu_0 ev_n)/(4pir_n^2)`
D. `(mu_0ev_n)/(4pi^2r_n^2)`

Explanation

To find the magnetic field at the nucleus of a hydrogen atom due to an electron revolving in the nth orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a hydrogen atom with an electron revolving in the nth orbit. The radius of this orbit is denoted as \( r_n \) and the velocity of the electron is \( v_n \).

2. **Current Representation**:
- The revolving electron can be treated as a current loop. The current \( I \) due to the electron can be defined as the charge flowing per unit time. The charge of the electron is \( e \).

3. **Calculating the Time Period**:
- The time \( T \) taken for the electron to complete one full revolution is given by the circumference of the orbit divided by the velocity of the electron:
\[ T = \frac{2 \pi r_n}{v_n} \]

4. **Calculating the Current**:
- The current \( I \) can be calculated using the formula:
\[ I = \frac{\text{Charge}}{\text{Time}} = \frac{e}{T} = \frac{e}{\frac{2 \pi r_n}{v_n}} = \frac{e v_n}{2 \pi r_n} \]

5. **Magnetic Field at the Center**:
- The magnetic field \( B \) at the center of a circular loop of radius \( r_n \) carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 r_n} \]
- Substituting the expression for \( I \):
\[ B = \frac{\mu_0 \left(\frac{e v_n}{2 \pi r_n}\right)}{2 r_n} = \frac{\mu_0 e v_n}{4 \pi r_n^2} \]

6. **Final Expression**:
- Therefore, the magnetic field at the nucleus of the hydrogen atom due to the revolving electron is:
\[ B = \frac{\mu_0 e v_n}{4 \pi r_n^2} \]