Bohr magnetonis given by (symbols have their usual meanings)

Options

A. `(4pi m _(e))/(eh^(2))`
B. `(4pi m _(e))/(eh)`
C. `(e h ^(2))/(4 pi m _(e))`
D. `(eh )/(4pi m _(e))`

Answer

To derive the expression for the Bohr magneton, we start from the fundamental definitions and relationships in physics. The Bohr magneton (\( \mu_B \)) is a physical constant that describes the magnetic moment of an electron due to its orbital or spin angular momentum. Here’s a step-by-step solution:

### Step 1: Understand the Definition
The Bohr magneton is defined as:
\[ \mu_B = \frac{e \cdot h}{4 \pi m} \]
where:
- \( e \) is the charge of the electron (\( 1.6 \times 10^{-19} \) coulombs),
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \) Js),
- \( m \) is the mass of the electron (\( 9.11 \times 10^{-31} \) kg).

### Step 2: Substitute the Values
Now, we will substitute the known values into the equation:
\[ \mu_B = \frac{(1.6 \times 10^{-19}) \cdot (6.626 \times 10^{-34})}{4 \pi (9.11 \times 10^{-31})} \]

### Step 3: Calculate the Denominator
First, calculate the denominator:
\[ 4 \pi m = 4 \pi (9.11 \times 10^{-31}) \approx 1.143 \times 10^{-30} \text{ kg} \]

### Step 4: Calculate the Numerator
Next, calculate the numerator:
\[ e \cdot h = (1.6 \times 10^{-19}) \cdot (6.626 \times 10^{-34}) \approx 1.059 \times 10^{-52} \text{ J s} \]

### Step 5: Divide the Numerator by the Denominator
Now, divide the numerator by the denominator:
\[ \mu_B = \frac{1.059 \times 10^{-52}}{1.143 \times 10^{-30}} \approx 9.27 \times 10^{-24} \text{ J/T} \]

### Step 6: Final Result
Thus, the value of the Bohr magneton is:
\[ \mu_B \approx 9.27 \times 10^{-24} \text{ J/T} \]

### Conclusion
The Bohr magneton is a fundamental physical constant that quantifies the magnetic moment of an electron due to its angular momentum.