If an electron travels with a velocity of 1/100th speed of light in the first in the first Bohr orbit, what is its velocity (relative to the speed of light) in the 5th Bohr orbit ?

Options

A. 0.002
B. 0.1
C. 0.5
D. 0.7

Explanation

To solve the problem, we need to determine the velocity of an electron in the 5th Bohr orbit given its velocity in the 1st Bohr orbit. We will use the formula for the velocity of an electron in a Bohr orbit, which is given by:

\[ V_n = V_0 \cdot \frac{Z}{n} \]

where:
- \( V_n \) is the velocity of the electron in the nth orbit,
- \( V_0 \) is the velocity of the electron in the first orbit,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)),
- \( n \) is the principal quantum number (the orbit number).

### Step-by-Step Solution:

1. **Identify the velocity in the first orbit**:
The problem states that the electron travels with a velocity of \( \frac{1}{100} \) the speed of light in the first Bohr orbit. Therefore:
\[ V_1 = \frac{C}{100} \]
where \( C \) is the speed of light.

2. **Determine \( V_0 \)**:
From the formula, we know that:
\[ V_0 = V_1 = \frac{C}{100} \]

3. **Calculate the velocity in the 5th orbit**:
Now, we need to find the velocity in the 5th Bohr orbit (\( n = 5 \)):
\[ V_5 = V_0 \cdot \frac{Z}{5} \]
Since we are considering hydrogen (where \( Z = 1 \)):
\[ V_5 = \left(\frac{C}{100}\right) \cdot \frac{1}{5} \]

4. **Simplify the expression**:
\[ V_5 = \frac{C}{100} \cdot \frac{1}{5} = \frac{C}{500} \]

5. **Express \( V_5 \) as a fraction of the speed of light**:
Thus, the velocity of the electron in the 5th Bohr orbit is:
\[ V_5 = 0.002C \]

### Final Answer:
The velocity of the electron in the 5th Bohr orbit is \( 0.002C \).