According to Bohr model, magnetic field at the centre (at the nucleus) of a hydrogen atom due to the motion of the electron in nth orbit is proportional to `1//n^(x)`, find the value of x

Answer

To find the value of \( x \) in the expression for the magnetic field at the center of a hydrogen atom according to the Bohr model, we will follow these steps:

### Step 1: Understand the magnetic field due to a current loop
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) 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 loop.

### Step 2: Identify the current in the nth orbit
In the Bohr model, the electron moving in the nth orbit can be considered as a current loop. The current \( I_n \) due to the electron in the nth orbit can be expressed as:
\[ I_n = \frac{e}{T_n} \]
where \( e \) is the charge of the electron and \( T_n \) is the time period of the electron in the nth orbit.

### Step 3: Relate the time period to the orbit number
The time period \( T_n \) can be related to the radius of the orbit \( R_n \) and the speed \( v_n \) of the electron. According to Bohr's model, the radius of the nth orbit is given by:
\[ R_n \propto n^2 \]
The speed of the electron \( v_n \) is proportional to \( \frac{1}{n} \). Therefore, the time period \( T_n \) can be expressed as:
\[ T_n \propto \frac{R_n}{v_n} \propto \frac{n^2}{\frac{1}{n}} = n^3 \]

### Step 4: Substitute \( T_n \) into the current expression
Now substituting \( T_n \) into the current expression:
\[ I_n \propto \frac{e}{n^3} \]

### Step 5: Determine the radius of the nth orbit
From the previous steps, we have:
\[ R_n \propto n^2 \]

### Step 6: Substitute \( I_n \) and \( R_n \) into the magnetic field formula
Now substituting \( I_n \) and \( R_n \) into the magnetic field formula:
\[ B \propto \frac{\mu_0 I_n}{2R_n} \propto \frac{\mu_0 \left(\frac{e}{n^3}\right)}{2(n^2)} = \frac{\mu_0 e}{2n^5} \]

### Step 7: Express the magnetic field in terms of \( n \)
From the above expression, we can see that:
\[ B \propto \frac{1}{n^5} \]

### Step 8: Identify the value of \( x \)
The question states that the magnetic field \( B \) is proportional to \( \frac{1}{n^x} \). From our derivation, we have:
\[ B \propto \frac{1}{n^5} \]
Thus, we can conclude that:
\[ x = 5 \]

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

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