The electron in the ground state of hydrogen atom produces a magnetic field `B` ạt the nucleus. This magnetic field' depends on the value of many constants. If `e` is electronic charge, `B` is found to be proportional to `e^(n) .` Find `n`.

Answer

To find the value of \( n \) in the expression \( B \propto e^n \) for the magnetic field \( B \) produced by the electron in the ground state of a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Magnetic Field**:
The magnetic field \( B \) at the nucleus due to the electron can be expressed in terms of the current \( i \) produced by the electron's motion. The formula for the magnetic field at the center of a circular loop is given by:
\[ B = \frac{\mu_0 i}{2r} \]
where \( \mu_0 \) is the permeability of free space, \( i \) is the current, and \( r \) is the radius of the electron's orbit.

2. **Expressing Current**:
The current \( i \) due to the electron can be expressed as:
\[ i = \frac{e}{T} \]
where \( e \) is the electronic charge and \( T \) is the time period of the electron's orbit. The time period \( T \) can be expressed in terms of the radius \( r \) and the velocity \( v \) of the electron:
\[ T = \frac{2\pi r}{v} \]
Thus, we can rewrite the current as:
\[ i = \frac{e v}{2\pi r} \]

3. **Substituting Current into Magnetic Field**:
Substituting the expression for current into the magnetic field formula gives:
\[ B = \frac{\mu_0 \left( \frac{e v}{2\pi r} \right)}{2r} = \frac{\mu_0 e v}{4\pi r^2} \]

4. **Finding the Relationships**:
We need to express \( v \) and \( r \) in terms of \( e \) and other constants. For an electron in a hydrogen atom, we can use the Bohr model:
- The velocity \( v \) of the electron is given by:
\[ v \propto \frac{e^2 k}{n h} \]
where \( k \) is Coulomb's constant, \( n \) is the principal quantum number, and \( h \) is Planck's constant.
- The radius \( r \) of the electron's orbit is given by:
\[ r \propto \frac{n^2 h^2}{4 \pi^2 m k e^2} \]

5. **Substituting \( v \) and \( r \) into \( B \)**:
Now substituting these relationships into the expression for \( B \):
\[ B \propto \frac{e \left( \frac{e^2 k}{n h} \right)}{r^2} \]
Since \( r \propto \frac{n^2 h^2}{4 \pi^2 m k e^2} \), we have:
\[ r^2 \propto \left( \frac{n^2 h^2}{4 \pi^2 m k e^2} \right)^2 \]
Therefore:
\[ B \propto \frac{e \cdot \frac{e^2 k}{n h}}{\left( \frac{n^2 h^2}{4 \pi^2 m k e^2} \right)^2} \]

6. **Simplifying the Expression**:
After simplifying the above expression, we find that:
\[ B \propto e^7 \]
Thus, we conclude that:
\[ n = 7 \]

### Final Answer:
The value of \( n \) is \( 7 \).