The magnetic fieold at the centre of a hydrogen atom due to the motion of the electron in the first Bohr orbit is B . The magnetic field at the centre due to the motion of the electron in the second Bohr orbit will be `(B)/(2^(x))`Find value of x.

Answer

To find the value of \( x \) in the expression for the magnetic field at the center of a hydrogen atom due to the motion of the electron in the second Bohr orbit, we can follow these steps:

### Step 1: Understand the relationship between magnetic field and electron motion
The magnetic field \( B \) at the center of a circular loop (like the orbit of an electron) is given by the formula:
\[ B \propto \frac{I}{R} \]
where \( I \) is the current and \( R \) is the radius of the orbit.

### Step 2: Determine the current due to the electron
The current \( I \) due to the 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 one complete revolution.

### Step 3: Relate the time period to the radius
The time period \( T \) can be expressed in terms of the velocity \( v_n \) and radius \( R_n \) of the orbit:
\[ T = \frac{2\pi R_n}{v_n} \]
Thus, the current can also be expressed as:
\[ I = \frac{e v_n}{2\pi R_n} \]

### Step 4: Analyze the radius and velocity in Bohr's model
From Bohr's model, we know:
- The radius of the \( n \)-th orbit is given by:
\[ R_n \propto n^2 \]
- The velocity \( v_n \) is inversely proportional to \( n \):
\[ v_n \propto \frac{1}{n} \]

### Step 5: Substitute the relationships into the magnetic field formula
Substituting \( I \) and \( R \) into the magnetic field equation, we get:
\[ B \propto \frac{e v_n}{2\pi R_n} \cdot \frac{1}{R_n} \]
This leads to:
\[ B \propto \frac{e v_n}{2\pi R_n^2} \]

### Step 6: Combine the relationships
Since \( v_n \propto \frac{1}{n} \) and \( R_n \propto n^2 \), we have:
\[ B_n \propto \frac{1/n}{(n^2)^2} = \frac{1}{n^5} \]
Thus, the magnetic field for the first orbit \( B_1 \) and the second orbit \( B_2 \) can be expressed as:
\[ B_1 \propto \frac{1}{1^5} = 1 \]
\[ B_2 \propto \frac{1}{2^5} = \frac{1}{32} \]

### Step 7: Relate \( B_2 \) to \( B_1 \)
From the above, we find:
\[ B_2 = \frac{B_1}{32} \]
Since \( B_1 = B \), we can write:
\[ B_2 = \frac{B}{2^5} \]

### Step 8: Identify \( x \)
From the expression \( B_2 = \frac{B}{2^x} \), we can see that \( x = 5 \).

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