The ratio of the speed of the electron in the first Bohr orbit of hydrogen and the speed of light is equal to (where `e, h and c` have their usual meaning in cgs system)

Options

A. `2 pi h // e^(2)`
B. `e r^(2) h// 2 pi c`
C. `e ^(2) c// 2 pi h`
D. `2 pi e^(2)// h c`

Explanation

To solve the problem of finding the ratio of the speed of the electron in the first Bohr orbit of hydrogen to the speed of light, we can follow these steps:

### Step 1: Understand the Bohr Model
In the Bohr model of the hydrogen atom, the speed of an electron in the nth orbit is given by the formula:
\[ v_n = \frac{k \cdot e^2}{n \cdot h} \]
where:
- \( k \) is the Coulomb's constant,
- \( e \) is the charge of the electron,
- \( n \) is the principal quantum number (for the first orbit, \( n = 1 \)),
- \( h \) is Planck's constant.

### Step 2: Substitute Values for the First Orbit
For the first Bohr orbit (where \( n = 1 \)):
\[ v_1 = \frac{k \cdot e^2}{1 \cdot h} = \frac{k \cdot e^2}{h} \]

### Step 3: Express Coulomb's Constant in CGS Units
In the CGS system, the value of Coulomb's constant \( k \) is given by:
\[ k = \frac{1}{4 \pi \epsilon_0} \]
In CGS units, \( \epsilon_0 = \frac{1}{4 \pi} \), thus:
\[ k = 1 \]
Therefore, we can simplify the equation for \( v_1 \):
\[ v_1 = \frac{e^2}{h} \]

### Step 4: Calculate the Ratio of Speed of Electron to Speed of Light
Now, we need to find the ratio of the speed of the electron to the speed of light \( c \):
\[ \frac{v_1}{c} = \frac{e^2/h}{c} = \frac{e^2}{h \cdot c} \]

### Step 5: Final Expression
Thus, the ratio of the speed of the electron in the first Bohr orbit of hydrogen to the speed of light is:
\[ \frac{v_1}{c} = \frac{e^2}{h \cdot c} \]

### Step 6: Simplifying the Expression
To express this in a more recognizable form, we can multiply the numerator and denominator by \( 2\pi \):
\[ \frac{v_1}{c} = \frac{2\pi e^2}{2\pi h \cdot c} = \frac{2\pi e^2}{h \cdot c} \]

### Conclusion
Thus, the final answer is:
\[ \frac{v_1}{c} = \frac{2\pi e^2}{h \cdot c} \]