A standard 50W electirc bulb in series with a room heater is connected across the mains. If the 50W bulb is replaced by `100W` bulb the heater output will

Options

A. Double
B. Halve
C. increase
D. Decrease

Answer

To solve the problem, we need to analyze the effect of replacing a 50W bulb with a 100W bulb in a series circuit with a room heater.

### Step-by-Step Solution:

1. **Understanding the Circuit**:
- We have a series circuit consisting of a 50W bulb and a room heater connected across the mains voltage (let's denote this voltage as \( E \)).
- The power rating of the bulbs indicates how much power they consume when connected to the mains.

2. **Power and Resistance Relationship**:
- The power \( P \) consumed by a device can be expressed as:
\[ P = \frac{V^2}{R} \]
- Where \( V \) is the voltage across the device and \( R \) is its resistance.
- For both bulbs, since they are connected to the same mains voltage, the voltage \( V \) remains constant.

3. **Calculating Resistance of Each Bulb**:
- For the 50W bulb:
\[ P_1 = 50W \implies R_1 = \frac{V^2}{P_1} \]
- For the 100W bulb:
\[ P_2 = 100W \implies R_2 = \frac{V^2}{P_2} \]

4. **Comparing Resistances**:
- Since \( P_1 < P_2 \), it follows that \( R_1 > R_2 \). This means that the resistance of the 50W bulb is greater than that of the 100W bulb.

5. **Net Resistance in the Circuit**:
- When the 50W bulb is replaced with the 100W bulb, the total resistance in the circuit decreases:
\[ R_{\text{net}} = R_{\text{bulb}} + R_{\text{heater}} \implies R_{\text{net}} \text{ decreases} \]

6. **Effect on Current**:
- According to Ohm's law, the current \( I \) in the circuit is given by:
\[ I = \frac{E}{R_{\text{net}}} \]
- Since \( R_{\text{net}} \) decreases, the current \( I \) in the circuit increases.

7. **Impact on Heater Output**:
- The heater's output power is given by:
\[ P_{\text{heater}} = I^2 R_{\text{heater}} \]
- With an increase in current \( I \), the power output of the heater will also increase.

### Conclusion:
When the 50W bulb is replaced by a 100W bulb, the output of the heater will increase.