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 understand the relationship between the direct current (DC) voltage and the alternating current (AC) voltage in terms of the heat produced by the heater.

### Step-by-Step Solution:

1. **Understand the Heat Produced by the Heater:**
The heat produced by an electrical device is given by the formula:
\[ H = I^2 R t \]
where \( H \) is the heat produced, \( I \) is the current, \( R \) is the resistance, and \( t \) is the time.

2. **Determine the Resistance of the Heater:**
When the heater is connected to a 110 V DC source, the current \( I_{DC} \) can be expressed as:
\[ I_{DC} = \frac{V_{DC}}{R} = \frac{110}{R} \]
Therefore, the heat produced when connected to a DC source is:
\[ H_{DC} = I_{DC}^2 R t = \left(\frac{110}{R}\right)^2 R t = \frac{110^2}{R} t \]

3. **Relate AC Voltage to Heat Production:**
When the heater is connected to an AC source with an RMS voltage \( V_{rms} \), the current \( I_{AC} \) can be expressed as:
\[ I_{AC} = \frac{V_{rms}}{R} \]
The heat produced by the AC source is:
\[ H_{AC} = I_{AC}^2 R t = \left(\frac{V_{rms}}{R}\right)^2 R t = \frac{V_{rms}^2}{R} t \]

4. **Set the Heat Produced by DC Equal to AC:**
Since the heat produced by both sources is the same:
\[ H_{DC} = H_{AC} \]
Substituting the expressions for heat:
\[ \frac{110^2}{R} t = \frac{V_{rms}^2}{R} t \]

5. **Cancel Common Terms:**
Since \( R \) and \( t \) are common on both sides, we can cancel them out:
\[ 110^2 = V_{rms}^2 \]

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

### Final Answer:
The RMS value of the alternating voltage is \( 110 \, \text{V} \).

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