two electron are revolving around a nucleus at distance `r and `4 r`. The ratio of their periods is

Options

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

Explanation

To solve the problem of finding the ratio of the periods of two electrons revolving around a nucleus at distances \( r \) and \( 4r \), we will use Kepler's third law, which states that the square of the period of revolution \( T \) is directly proportional to the cube of the radius \( r \) of the orbit.

### Step-by-Step Solution:

1. **Understand the relationship between period and radius**:
According to Kepler's third law, we have:
\[ T^2 \propto r^3 \]
This means that:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

2. **Assign the distances**:
Let \( r_1 = r \) (the distance of the first electron) and \( r_2 = 4r \) (the distance of the second electron).

3. **Substitute the values into the ratio**:
We can now write:
\[ \frac{T_r^2}{T_{4r}^2} = \frac{r^3}{(4r)^3} \]

4. **Calculate the right-hand side**:
Simplifying the right-hand side:
\[ (4r)^3 = 64r^3 \]
Therefore, we have:
\[ \frac{T_r^2}{T_{4r}^2} = \frac{r^3}{64r^3} = \frac{1}{64} \]

5. **Take the square root to find the ratio of periods**:
Taking the square root of both sides gives:
\[ \frac{T_r}{T_{4r}} = \frac{1}{8} \]

6. **Express the ratio**:
Thus, the ratio of the periods \( T_r : T_{4r} \) is:
\[ T_r : T_{4r} = 1 : 8 \]

### Final Answer:
The ratio of the periods of the two electrons is \( 1 : 8 \).