A coil of self-inductance `L` is connected in series with a bulb `B` and an `AC` source. Brightness of the bulb decreases when

Options

A. (a) frequency of the AC source is decreased
B. (b) number of turns in hte coil is reduced
C. (c) a capacitance of reactance `X_(C) = X_(L)` is included in the same circuit.
D. (d) an iron rod is inserted in the coil

Explanation

To solve the problem, we need to analyze how the self-inductance of a coil affects the brightness of a bulb when connected in series with an AC source. The brightness of the bulb is related to the power consumed in the circuit, which depends on the current flowing through it.

### Step-by-Step Solution:

1. **Understand the Circuit**: The circuit consists of a coil (inductor) with self-inductance \( L \), a bulb (which has resistance \( R \)), and an AC source. The components are connected in series.

2. **Power and Brightness Relationship**: The brightness of the bulb is related to the power \( P \) consumed in the circuit. The power can be expressed as:
\[ P = I^2 R \]
where \( I \) is the current flowing through the circuit.

3. **Current in the Circuit**: The current \( I \) in an AC circuit with an inductor is given by the formula:
\[ I = \frac{E}{Z} \]
where \( E \) is the EMF of the AC source and \( Z \) is the impedance of the circuit.

4. **Impedance Calculation**: The impedance \( Z \) in a series circuit with resistance \( R \) and inductive reactance \( X_L \) is given by:
\[ Z = \sqrt{R^2 + X_L^2} \]
where \( X_L = L \omega \) and \( \omega = 2 \pi f \) (with \( f \) being the frequency of the AC source).

5. **Condition for Decreased Brightness**: For the brightness of the bulb to decrease, the current \( I \) must decrease. This can happen if the impedance \( Z \) increases.

6. **Increasing Impedance**: The impedance \( Z \) can increase in two ways:
- By increasing the inductance \( L \).
- By increasing the frequency \( f \) of the AC source.

7. **Analyze the Options**:
- **Option A**: Decreasing the frequency of the AC source would decrease \( X_L \) and thus decrease \( Z \). This would not decrease the brightness.
- **Option B**: Reducing the number of turns in the coil would decrease \( L \), thus decreasing \( X_L \) and \( Z \). This would not decrease the brightness.
- **Option C**: When \( X_C = X_L \) (resonance condition), the impedance is minimized. This would not decrease the brightness.
- **Option D**: Inserting an iron rod into the coil increases the inductance \( L \), which increases \( X_L \) and thus increases \( Z \). This would decrease the brightness.

8. **Conclusion**: The correct answer is **Option D**: An iron rod is inserted in the coil, which increases the inductance \( L \) and thus decreases the brightness of the bulb.