Assertion (A) : An electric lamp is connected in series with a long solenoid of copper with air core and then connected to `AC` source. If an iron rod is inserted in solenoid the lamp will become dim.
Reason (R ) : If iron rod is inserted in solenoid, the induction of solenoid increases.

Options

A. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
B. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true

Explanation

To solve the problem, we need to analyze the assertion and reason provided.

### Step-by-Step Solution:

1. **Understanding the Assertion (A)**:
- The assertion states that when an electric lamp is connected in series with a long solenoid (air core) and an AC source, inserting an iron rod into the solenoid will cause the lamp to become dim.
- The lamp becoming dim indicates that the current flowing through the circuit has decreased.

2. **Understanding the Reason (R)**:
- The reason states that inserting an iron rod into the solenoid increases the inductance of the solenoid.
- Inductance (L) is a property of the solenoid that opposes changes in current. When the inductance increases, the reactance (X_L) also increases, which affects the current in the circuit.

3. **Analyzing Inductance**:
- The inductance of a solenoid is given by the formula:
\[ L = \mu_r \cdot \mu_0 \cdot \frac{N^2 \cdot A}{l} \]
where:
- \( \mu_r \) is the relative permeability of the core material (which increases when an iron rod is inserted),
- \( \mu_0 \) is the permeability of free space,
- \( N \) is the number of turns,
- \( A \) is the cross-sectional area,
- \( l \) is the length of the solenoid.

4. **Effect of Inserting Iron Rod**:
- When the iron rod is inserted, \( \mu_r \) increases, which leads to an increase in \( L \).
- An increase in inductance results in an increase in inductive reactance \( X_L \) given by:
\[ X_L = \omega L \]
where \( \omega \) is the angular frequency of the AC source.

5. **Current and Lamp Brightness**:
- The current \( I \) in the circuit can be expressed as:
\[ I = \frac{V}{X_L} \]
where \( V \) is the voltage of the AC source.
- Since \( X_L \) increases due to the increase in inductance, the current \( I \) decreases.
- A decrease in current results in the lamp becoming dimmer.

6. **Conclusion**:
- The assertion is true: the lamp will become dim when the iron rod is inserted.
- The reason is also true: inserting the iron rod increases the inductance of the solenoid.
- Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

### Final Answer:
Both Assertion (A) and Reason (R) are true, and R explains A.

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