What is the correct value of Bohr magneton?

Options

A. `8.89 xx 10^(-24)A m^(2)`
B. `9.27 xx 10^(-24)A m^(2)`
C. `5.56 xx 10^(-24)A m^(2)`
D. `9.27 xx 10^(-28)A m^(2)`

Explanation

To find the correct value of the Bohr magneton, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The Bohr magneton (μ_B) is given by the formula:
\[ \mu_B = \frac{e \cdot h}{4 \pi m_e} \]
where:
- \( e \) is the charge of the electron,
- \( h \) is Planck's constant,
- \( m_e \) is the mass of the electron.

2. **Substitute the Known Values**:
- \( e = 1.6 \times 10^{-19} \, \text{C} \)
- \( h = 6.63 \times 10^{-34} \, \text{J s} \) (Note: The value in the transcript is slightly different, but this is a more commonly accepted value)
- \( m_e = 9.1 \times 10^{-31} \, \text{kg} \)

Plugging these values into the formula:
\[ \mu_B = \frac{(1.6 \times 10^{-19}) \cdot (6.63 \times 10^{-34})}{4 \pi (9.1 \times 10^{-31})} \]

3. **Calculate the Denominator**:
- First, calculate \( 4 \pi m_e \):
\[ 4 \pi (9.1 \times 10^{-31}) \approx 1.143 \times 10^{-30} \]

4. **Calculate the Numerator**:
- Now calculate \( e \cdot h \):
\[ (1.6 \times 10^{-19}) \cdot (6.63 \times 10^{-34}) \approx 1.061 \times 10^{-52} \]

5. **Final Calculation**:
- Now divide the numerator by the denominator:
\[ \mu_B = \frac{1.061 \times 10^{-52}}{1.143 \times 10^{-30}} \approx 9.27 \times 10^{-24} \, \text{A m}^2 \]

6. **Conclusion**:
- Therefore, the correct value of the Bohr magneton is:
\[ \mu_B \approx 9.27 \times 10^{-24} \, \text{A m}^2 \]