The gyro-magnetic ratio of an electron in an H-atom, according to Bohr model, is

Options

A. independent of which orbit it is in
B. negative
C. positive
D. increases with the quantum number n.

Explanation

To find the gyromagnetic ratio of an electron in a hydrogen atom according to the Bohr model, we will follow these steps:

### Step 1: Understand the Gyromagnetic Ratio
The gyromagnetic ratio (γ) is defined as the ratio of the magnetic moment (μ) to the angular momentum (L) of the electron. Mathematically, it can be expressed as:
\[ \gamma = \frac{\mu}{L} \]

### Step 2: Calculate Angular Momentum (L)
According to Bohr's model, the angular momentum (L) of an electron in a hydrogen atom is quantized and given by:
\[ L = mvr = \frac{nh}{2\pi} \]
where:
- \( m \) is the mass of the electron,
- \( v \) is the speed of the electron,
- \( r \) is the radius of the orbit,
- \( n \) is the principal quantum number,
- \( h \) is Planck's constant.

### Step 3: Calculate Magnetic Moment (μ)
The magnetic moment (μ) for a current loop can be expressed as:
\[ \mu = I \cdot A \]
where:
- \( I \) is the current,
- \( A \) is the area of the loop.

For a single electron moving in a circular orbit, the current \( I \) can be defined as:
\[ I = \frac{q}{T} \]
where \( q \) is the charge of the electron and \( T \) is the time period of the orbit. The area \( A \) of the circular orbit is:
\[ A = \pi r^2 \]

The time period \( T \) can be expressed as:
\[ T = \frac{2\pi r}{v} \]
Thus, substituting \( T \) into the expression for current, we have:
\[ I = \frac{qv}{2\pi r} \]

Now substituting \( I \) into the expression for magnetic moment:
\[ \mu = \left(\frac{qv}{2\pi r}\right) \cdot \pi r^2 = \frac{qvr}{2} \]

### Step 4: Substitute into Gyromagnetic Ratio Formula
Now we can substitute the expressions for \( \mu \) and \( L \) into the gyromagnetic ratio formula:
\[ \gamma = \frac{\mu}{L} = \frac{\frac{qvr}{2}}{\frac{nh}{2\pi}} = \frac{qvr \cdot 2\pi}{2nh} = \frac{qv \cdot \pi r}{nh} \]

### Step 5: Simplifying the Gyromagnetic Ratio
We know that \( L = mvr \), hence:
\[ \gamma = \frac{q}{2m} \]

### Step 6: Conclusion
The gyromagnetic ratio of an electron in a hydrogen atom according to the Bohr model is:
\[ \gamma = \frac{e}{2m} \]
where \( e \) is the charge of the electron and \( m \) is the mass of the electron. This ratio is a constant value and does not depend on the quantum number \( n \).