Single electron is orbiting in `n^(th)`orbit of hydrogen atom. The magnetic field produced by the electron at the nucleus will be proportional to `n^(-k)` . Find the value of k.

Answer

To find the value of \( k \) in the expression for the magnetic field produced by an electron in the \( n^{th} \) orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the Magnetic Field at the Nucleus
The magnetic field \( B \) at the center of a circular current loop is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( R \) is the radius of the circular path.

### Step 2: Determine the Current Due to the Electron
The current \( I \) due to an electron moving in a circular orbit can be expressed as:
\[ I = \frac{e}{T} \]
where \( e \) is the charge of the electron and \( T \) is the time period of the electron's orbit. The time period \( T \) for an electron in the \( n^{th} \) orbit is given by:
\[ T = \frac{2\pi r_n}{v_n} \]
where \( r_n \) is the radius of the \( n^{th} \) orbit and \( v_n \) is the velocity of the electron in that orbit.

### Step 3: Substitute the Time Period into the Current Expression
Substituting \( T \) into the current expression gives:
\[ I = \frac{e v_n}{2\pi r_n} \]

### Step 4: Substitute Current into the Magnetic Field Formula
Now, substituting the expression for current \( I \) into the magnetic field formula:
\[ B = \frac{\mu_0}{2R} \cdot \frac{e v_n}{2\pi r_n} \]

### Step 5: Analyze the Relationships of Velocity and Radius
From Bohr's model of the hydrogen atom:
- The radius \( r_n \) of the \( n^{th} \) orbit is proportional to \( n^2 \):
\[ r_n \propto n^2 \]
- The velocity \( v_n \) of the electron in the \( n^{th} \) orbit is inversely proportional to \( n \):
\[ v_n \propto \frac{1}{n} \]

### Step 6: Substitute the Proportional Relationships
Substituting these relationships into the expression for \( B \):
\[ B \propto \frac{v_n}{r_n^2} \propto \frac{1/n}{(n^2)^2} = \frac{1/n}{n^4} = \frac{1}{n^5} \]

### Step 7: Determine the Value of \( k \)
From the expression \( B \propto n^{-5} \), we can see that:
\[ k = 5 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{5} \]