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 solve the problem, we need to understand how the magnetic field at the center of a hydrogen atom changes with the electron's orbit.

### Step-by-Step Solution:

1. **Understanding the Magnetic Field (B)**:
The magnetic field at the center of a hydrogen atom due to the motion of an electron can be expressed using the formula for the magnetic field created by a current loop. The magnetic field \( B \) at the center of a circular loop of radius \( r \) carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space.

2. **Identifying Current (I)**:
The current \( I \) due to the electron can be defined as:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge of the electron and \( T \) is the time period for one complete revolution.

3. **Relating Time Period (T) and Radius (r)**:
For Bohr orbits, the radius \( r \) is proportional to \( n^2 \) (where \( n \) is the principal quantum number) and the time period \( T \) is proportional to \( n^3 \). Therefore, we can express:
\[ r \propto n^2 \quad \text{and} \quad T \propto n^3 \]

4. **Substituting in the Magnetic Field Formula**:
From the above relationships, we can rewrite the magnetic field \( B \) as:
\[ B \propto \frac{I}{r} \propto \frac{Q/T}{r} \propto \frac{Q}{n^3 \cdot n^2} = \frac{Q}{n^5} \]
This shows that the magnetic field \( B \) is inversely proportional to \( n^5 \).

5. **Calculating Magnetic Field for Different Orbits**:
For the first orbit (n=1), we denote the magnetic field as \( B_1 = B \). For the second orbit (n=2):
\[ B_2 \propto \frac{1}{2^5} = \frac{1}{32} \]
Thus, we can express \( B_2 \) in terms of \( B \):
\[ B_2 = \frac{B}{2^5} \]

6. **Identifying the Value of x**:
From the expression \( B_2 = \frac{B}{2^x} \), we can see that:
\[ x = 5 \]

### Final Answer:
The value of \( x \) is **5**.