Gyromagnetic ratio is the ratio of magnetic moment `(mu_(l))` to the orbital angular momentum (l). Its numerical value for an electron is given by

Options

A. `8.8xx 10^(-12)C" kg "^(-1)`
B. `8.8xx 10^(10)C" kg "^(-1)`
C. `1.6 xx 10^(-19)C" kg "^(-1)`
D. `6.67 xx 10^(11)C" kg "^(-1)`

Explanation

To find the gyromagnetic ratio for an electron, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The gyromagnetic ratio (γ) is defined as the ratio of the magnetic moment (μ) to the orbital angular momentum (L). Mathematically, it is expressed as:
\[ \gamma = \frac{\mu}{L} \]

2. **Use the Known Relation**: For any charged particle, the gyromagnetic ratio can also be expressed as:
\[ \gamma = \frac{e}{2m} \]
where:
- \( e \) is the charge of the particle,
- \( m \) is the mass of the particle.

3. **Substitute Values for an Electron**:
- The charge of an electron \( e = 1.6 \times 10^{-19} \) coulombs.
- The mass of an electron \( m = 9.1 \times 10^{-31} \) kg.

4. **Calculate the Gyromagnetic Ratio**:
Substitute the values into the formula:
\[ \gamma = \frac{1.6 \times 10^{-19}}{2 \times 9.1 \times 10^{-31}} \]

5. **Perform the Calculation**:
- First, calculate the denominator:
\[ 2 \times 9.1 \times 10^{-31} = 1.82 \times 10^{-30} \]
- Now, divide the charge by this value:
\[ \gamma = \frac{1.6 \times 10^{-19}}{1.82 \times 10^{-30}} \approx 8.84 \times 10^{10} \text{ C kg}^{-1} \]

6. **Final Result**: The numerical value of the gyromagnetic ratio for an electron is approximately:
\[ \gamma \approx 8.84 \times 10^{10} \text{ C kg}^{-1} \]