Product of velocity and time period of electron orbiting in nth stable orbit is proportional to

Options

A. `n^(3)`
B. `(1)/(n)`
C. n
D. `n^(2)`

Explanation

To solve the question, we need to analyze the relationship between the velocity and time period of an electron in the nth stable orbit.

### Step-by-Step Solution:

1. **Understand the Concepts**:
- The velocity (v) of an electron in the nth orbit is given by the formula:
\[ v \propto \frac{1}{n} \]
- The time period (T) of the electron in the nth orbit is given by the formula:
\[ T \propto n^3 \]

2. **Express the Product**:
- We need to find the product of velocity and time period:
\[ v \cdot T \]

3. **Substituting the Proportional Relationships**:
- Substitute the expressions for velocity and time period into the product:
\[ v \cdot T \propto \left(\frac{1}{n}\right) \cdot (n^3) \]

4. **Simplifying the Expression**:
- Simplify the product:
\[ v \cdot T \propto \frac{n^3}{n} = n^2 \]

5. **Conclusion**:
- Therefore, the product of the velocity and the time period of an electron orbiting in the nth stable orbit is proportional to \( n^2 \).

### Final Answer:
The product of velocity and time period of an electron orbiting in the nth stable orbit is proportional to \( n^2 \). Thus, the correct option is option 4: \( n^2 \).