An Ac voltage is represented by e=220 sin `(100pi)` t volt and is applied over a resistance of 110 ohm. Calculate the heat produced in 7 min.

Options

A. `11xx10^(3)`cal
B. `22xx10^(3)`cal
C. `33xx10^(3)`cal
D. `25xx10^(3)`cal

Answer

To solve the problem of calculating the heat produced in an AC circuit with the given parameters, we can follow these steps:

### Step 1: Identify the given parameters
- The AC voltage is given as \( e = 220 \sin(100\pi t) \) volts.
- The resistance \( R = 110 \, \Omega \).
- The time duration \( t = 7 \) minutes.

### Step 2: Convert time from minutes to seconds
Since the time is given in minutes, we need to convert it to seconds for our calculations:
\[ t = 7 \, \text{minutes} \times 60 \, \text{seconds/minute} = 420 \, \text{seconds} \]

### Step 3: Calculate the peak voltage \( V_0 \)
From the given equation of the AC voltage, we can identify the peak voltage:
\[ V_0 = 220 \, \text{volts} \]

### Step 4: Calculate the RMS voltage \( V_{rms} \)
The RMS (Root Mean Square) voltage for an AC circuit is calculated using the formula:
\[ V_{rms} = \frac{V_0}{\sqrt{2}} = \frac{220}{\sqrt{2}} \approx 155.56 \, \text{volts} \]

### Step 5: Use the formula for heat produced
The heat produced in an electrical circuit can be calculated using the formula:
\[ H = \frac{V_{rms}^2}{R} \times t \]
Substituting the values we have:
\[ H = \frac{(155.56)^2}{110} \times 420 \]

### Step 6: Calculate \( H \)
First, calculate \( (155.56)^2 \):
\[ (155.56)^2 \approx 24167.78 \]
Now substitute this value into the heat formula:
\[ H = \frac{24167.78}{110} \times 420 \]
Calculating \( \frac{24167.78}{110} \):
\[ \frac{24167.78}{110} \approx 219.7 \]
Now, multiply by 420:
\[ H \approx 219.7 \times 420 \approx 92400 \, \text{joules} \]

### Step 7: Convert joules to calories
To convert joules to calories, we use the conversion factor \( 1 \, \text{calorie} = 4.2 \, \text{joules} \):
\[ H \approx \frac{92400}{4.2} \approx 22000 \, \text{calories} \]

### Final Answer
The heat produced in 7 minutes is approximately \( 22000 \, \text{calories} \).

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