In an atom, two electrons move around nucleus in circular orbits of radii ( R) and ( 4R) . The ratio of the time taken by them to complete one revolution is :

Options

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

Explanation

To solve the problem of finding the ratio of the time taken by two electrons to complete one revolution in circular orbits of radii \( R \) and \( 4R \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii**:
Let the radius of the first electron's orbit be \( r_1 = R \) and the radius of the second electron's orbit be \( r_2 = 4R \).

2. **Use Bohr's Model Relations**:
According to Bohr's model of the atom, the radius of the orbit is proportional to the square of the principal quantum number \( n \) (i.e., \( r \propto n^2 \)). The time period \( T \) (time taken to complete one revolution) is proportional to the cube of the principal quantum number \( n \) (i.e., \( T \propto n^3 \)).

3. **Establish the Relationship**:
From the above relations, we can express the time periods in terms of the radii:
\[ T_1 \propto n_1^3 \quad \text{and} \quad T_2 \propto n_2^3 \]
Since \( r_1 \propto n_1^2 \) and \( r_2 \propto n_2^2 \), we can relate the radii to the quantum numbers:
\[ \frac{r_1}{r_2} = \frac{n_1^2}{n_2^2} \]

4. **Find the Ratio of Time Periods**:
We can express the ratio of the time periods as:
\[ \frac{T_1}{T_2} = \left(\frac{r_1}{r_2}\right)^{3/2} \]
Substituting the values of \( r_1 \) and \( r_2 \):
\[ \frac{T_1}{T_2} = \left(\frac{R}{4R}\right)^{3/2} = \left(\frac{1}{4}\right)^{3/2} \]

5. **Calculate the Final Ratio**:
Simplifying the expression:
\[ \left(\frac{1}{4}\right)^{3/2} = \frac{1^3}{4^{3/2}} = \frac{1}{8} \]
Therefore, the ratio of the time periods is:
\[ \frac{T_1}{T_2} = \frac{1}{8} \]

6. **Express the Ratio**:
This can be expressed as:
\[ T_1 : T_2 = 1 : 8 \]

### Final Answer:
The ratio of the time taken by the two electrons to complete one revolution is \( 1 : 8 \).

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