A 110 V d.c. heater is used on an a.c source such that the heat produced is same as it produces when connected to 110 V dc in same time -intevals. What would be the r.m.s. value of the alternating voltage?

Options

A. 110 V
B. 220 V
C. 330 V
D. 440 V

Answer

To solve the problem, we need to determine the RMS (Root Mean Square) value of the alternating voltage that produces the same amount of heat as a 110 V DC heater.

### Step-by-Step Solution:

1. **Understanding the Heat Produced**:
The heat produced by an electrical device is proportional to the power consumed. For a resistive heater, the power (P) can be calculated using the formula:
\[ P = \frac{V^2}{R} \]
where \( V \) is the voltage and \( R \) is the resistance of the heater.

2. **Power in DC Circuit**:
For the DC heater connected to a 110 V source, the power can be expressed as:
\[ P_{DC} = \frac{(110)^2}{R} \]

3. **Power in AC Circuit**:
For the AC source, the RMS voltage is denoted as \( V_{rms} \). The power consumed by the heater when connected to the AC source can be expressed as:
\[ P_{AC} = \frac{(V_{rms})^2}{R} \]

4. **Setting the Powers Equal**:
Since the heat produced is the same in both cases, we can set the power equations equal to each other:
\[ P_{DC} = P_{AC} \]
This gives us:
\[ \frac{(110)^2}{R} = \frac{(V_{rms})^2}{R} \]

5. **Canceling Resistance**:
Since the resistance \( R \) is the same in both cases, we can cancel it out from both sides:
\[ (110)^2 = (V_{rms})^2 \]

6. **Solving for RMS Voltage**:
Taking the square root of both sides, we find:
\[ V_{rms} = 110 \text{ V} \]

### Conclusion:
The RMS value of the alternating voltage that produces the same heat as the 110 V DC heater is:
\[ \boxed{110 \text{ V}} \]