The ratio of the speed of electron in first Bohr orbit of H-atom to speed of light in vacuum is

Options

A. `137`
B. `7.30 xx 10^(-3)`
C. `100`
D. `10^(-2)`

Answer

To find the ratio of the speed of an electron in the first Bohr orbit of a hydrogen atom to the speed of light in vacuum, we can follow these steps:

### Step 1: Identify the speed of the electron in the first Bohr orbit
The speed of the electron in the first Bohr orbit of a hydrogen atom (n=1) is given by the formula:
\[ v_e = \frac{e^2}{2 \epsilon_0 h} \]
However, for simplicity, we can use the known value:
\[ v_e = 2.188 \times 10^8 \text{ cm/s} \]

### Step 2: Identify the speed of light in vacuum
The speed of light in vacuum is a constant:
\[ c = 3.0 \times 10^{10} \text{ cm/s} \]

### Step 3: Calculate the ratio of the speed of the electron to the speed of light
Now, we can calculate the ratio \( \frac{v_e}{c} \):
\[ \frac{v_e}{c} = \frac{2.188 \times 10^8 \text{ cm/s}}{3.0 \times 10^{10} \text{ cm/s}} \]

### Step 4: Perform the division
Now, we perform the division:
\[ \frac{v_e}{c} = \frac{2.188}{3.0} \times 10^{8 - 10} = \frac{2.188}{3.0} \times 10^{-2} \]
Calculating \( \frac{2.188}{3.0} \):
\[ \frac{2.188}{3.0} \approx 0.7293 \]
Thus,
\[ \frac{v_e}{c} \approx 0.7293 \times 10^{-2} = 7.293 \times 10^{-3} \]

### Step 5: Final answer
The ratio of the speed of the electron in the first Bohr orbit of the hydrogen atom to the speed of light in vacuum is approximately:
\[ \frac{v_e}{c} \approx 7.30 \times 10^{-3} \]

### Conclusion
The correct option based on the calculated ratio is option B.