In two H atoms A and B the electrons move around the nucleus in circular orbits of radius r and 4r respectively. The ratio of the times taken by them to complete one revolution is

Options

A. `1:4`
B. `1:2`
C. `1:8`
D. `2:1`

Explanation

To solve the problem of finding the ratio of the times taken by two hydrogen atoms A and B to complete one revolution, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii**:
- For atom A, the radius of the electron's orbit is \( r \).
- For atom B, the radius of the electron's orbit is \( 4r \).

2. **Understand the Motion**:
- The electrons in both atoms are moving in circular orbits. The time taken to complete one revolution depends on the circumference of the orbit and the speed of the electron.

3. **Calculate the Circumference**:
- The circumference \( C \) of a circular orbit is given by the formula:
\[ C = 2\pi r \]
- For atom A, the circumference \( C_1 \) is:
\[ C_1 = 2\pi r \]
- For atom B, the circumference \( C_2 \) is:
\[ C_2 = 2\pi (4r) = 8\pi r \]

4. **Determine the Speed of the Electrons**:
- The speed \( v \) of an electron in a circular orbit is inversely proportional to the square root of the radius of the orbit. Thus, we can express the speeds as:
\[ v \propto \frac{1}{\sqrt{r}} \]
- Therefore, for atom A (radius \( r \)):
\[ v_1 \propto \frac{1}{\sqrt{r}} \]
- And for atom B (radius \( 4r \)):
\[ v_2 \propto \frac{1}{\sqrt{4r}} = \frac{1}{2\sqrt{r}} \]

5. **Express the Speeds**:
- Let’s denote a constant \( k \) such that:
\[ v_1 = \frac{k}{\sqrt{r}} \quad \text{and} \quad v_2 = \frac{k}{2\sqrt{r}} \]

6. **Calculate the Times for One Revolution**:
- The time \( T \) taken to complete one revolution is given by:
\[ T = \frac{\text{Circumference}}{\text{Speed}} \]
- For atom A:
\[ T_1 = \frac{C_1}{v_1} = \frac{2\pi r}{\frac{k}{\sqrt{r}}} = \frac{2\pi r \sqrt{r}}{k} = \frac{2\pi r^{3/2}}{k} \]
- For atom B:
\[ T_2 = \frac{C_2}{v_2} = \frac{8\pi r}{\frac{k}{2\sqrt{r}}} = \frac{8\pi r \cdot 2\sqrt{r}}{k} = \frac{16\pi r^{3/2}}{k} \]

7. **Find the Ratio of Times**:
- Now, we can find the ratio \( \frac{T_1}{T_2} \):
\[ \frac{T_1}{T_2} = \frac{\frac{2\pi r^{3/2}}{k}}{\frac{16\pi r^{3/2}}{k}} = \frac{2}{16} = \frac{1}{8} \]

### Final Answer:
The ratio of the times taken by the electrons in atoms A and B to complete one revolution is:
\[ \frac{T_1}{T_2} = \frac{1}{8} \]