The magnetic field induction produced at the centre of orbit due to an electron revolving in `n^(th)` orbit of hydrogen atom is proportional to

Options

A. `n^(-3)`
B. `n^(-5)`
C. `n^(5)`
D. `n^(3)`

Answer

To find the magnetic field induction produced at the center of the orbit due to an electron revolving in the \( n^{th} \) orbit of a hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship of radius and velocity
The radius \( R \) of the \( n^{th} \) orbit in a hydrogen atom is given by:
\[ R \propto n^2 \]
The velocity \( V \) of the electron in the \( n^{th} \) orbit is given by:
\[ V \propto \frac{Z}{n} \]
For hydrogen, \( Z = 1 \), so:
\[ V \propto \frac{1}{n} \]

### Step 2: Determine the current due to the electron's motion
The current \( I \) due to the revolving electron can be expressed as:
\[ I = \frac{\text{Charge}}{\text{Time Period}} = \frac{e}{T} \]
where \( e \) is the charge of the electron and \( T \) is the time period of one complete revolution.

### Step 3: Calculate the time period \( T \)
The time period \( T \) can be calculated as:
\[ T = \frac{\text{Circumference}}{\text{Velocity}} = \frac{2\pi R}{V} \]
Substituting the expressions for \( R \) and \( V \):
\[ T \propto \frac{2\pi n^2}{\frac{1}{n}} = 2\pi n^3 \]

### Step 4: Substitute \( T \) back into the current equation
Now substituting \( T \) back into the current equation:
\[ I \propto \frac{e}{2\pi n^3} \]

### Step 5: Calculate the magnetic field \( B \)
The magnetic field \( B \) at the center of the orbit is given by:
\[ B = \frac{\mu_0 I}{2R} \]
Substituting the expressions for \( I \) and \( R \):
\[ B \propto \frac{\mu_0 \left(\frac{e}{2\pi n^3}\right)}{2(n^2)} \]
This simplifies to:
\[ B \propto \frac{\mu_0 e}{4\pi n^5} \]

### Step 6: Conclude the proportionality
From the above expression, we can conclude that:
\[ B \propto \frac{1}{n^5} \]
Thus, the magnetic field induction produced at the center of the orbit due to an electron revolving in the \( n^{th} \) orbit of a hydrogen atom is proportional to \( n^{-5} \).

### Final Answer
The magnetic field induction is proportional to \( n^{-5} \).

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