Statement I: An eletric lamp is connected in series with a long solenoid of copper with air core and then connected to an ac source. If an iron rod is inserted in the solenoid, the lamp will become dim.
Satement II: If an iron rod is inserted in the solenoid, the inductance of the solenoid increases.

Options

A. Statement I is True, Statement II is True, Statement II is the correct explanation for Statement I.
B. Statement I is True, Statement II is True, Statement II is NOT the correct explanation for Statement I.
C. Statement I is True, Statement II is False.
D. Statement I is False, Statement II is True.

Explanation

To solve the problem, we need to analyze the two statements provided regarding the behavior of an electric lamp connected in series with a solenoid when an iron rod is inserted into the solenoid.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have an electric lamp connected in series with a long solenoid (with an air core) to an AC source.
- The solenoid has a certain inductance which affects the current flowing through the circuit.

2. **Effect of Inserting an Iron Rod**:
- When an iron rod is inserted into the solenoid, the magnetic permeability of the core changes.
- The permeability of iron (μ_r) is greater than that of air (μ_0), which means the inductance of the solenoid will increase.

3. **Inductance and Inductive Reactance**:
- The self-inductance (L) of the solenoid can be expressed as:
\[ L = \frac{\mu n^2 A}{l} \]
where:
- μ is the permeability of the core,
- n is the number of turns per unit length,
- A is the cross-sectional area,
- l is the length of the solenoid.
- When the iron rod is inserted, the inductance becomes:
\[ L' = \frac{\mu_r n^2 A}{l} \]
Since μ_r > μ_0, we conclude that L' > L.

4. **Current in the Circuit**:
- The inductive reactance (X_L) is given by:
\[ X_L = \omega L \]
where ω is the angular frequency of the AC source.
- As the inductance increases (L' > L), the inductive reactance also increases (X_L' > X_L).

5. **Effect on Current**:
- The current (I) in the circuit can be expressed as:
\[ I = \frac{E}{X_L} \]
where E is the EMF of the AC source.
- Since X_L increases, the current I decreases.

6. **Conclusion on Lamp Brightness**:
- A decrease in current means that the electric lamp connected in series will become dimmer when the iron rod is inserted into the solenoid.

7. **Verification of Statements**:
- **Statement I**: The lamp will become dim. This is true because the current decreases due to increased inductance.
- **Statement II**: The inductance of the solenoid increases when an iron rod is inserted. This is also true as explained above.

8. **Final Conclusion**:
- Both statements are true, and Statement II correctly explains Statement I. Therefore, the correct answer is that both statements are true.