Three identical resistors are connected in series . When a certain potential difference is applied across the combination , the total power would be dissipated is `27 W`. How many times the power would be dissipated if the three resistors were connected in parallel across the same potential difference ?

Explanation

To solve the problem step by step, we will analyze the power dissipation in both series and parallel configurations of the resistors.

### Step 1: Understand the Series Configuration
When three identical resistors (each of resistance \( R \)) are connected in series, the total resistance \( R_{\text{eq, series}} \) is given by:
\[ R_{\text{eq, series}} = R + R + R = 3R \]

### Step 2: Calculate Power in Series
The power \( P \) dissipated in a resistor is given by the formula:
\[ P = \frac{V^2}{R_{\text{eq}}} \]
For the series configuration, substituting \( R_{\text{eq, series}} \):
\[ P_{\text{series}} = \frac{V^2}{3R} \]
According to the problem, the total power dissipated in the series configuration is \( 27 \, \text{W} \):
\[ \frac{V^2}{3R} = 27 \]
From this, we can express \( V^2 \) in terms of \( R \):
\[ V^2 = 27 \times 3R = 81R \]

### Step 3: Understand the Parallel Configuration
Now, when the same three resistors are connected in parallel, the equivalent resistance \( R_{\text{eq, parallel}} \) is given by:
\[ \frac{1}{R_{\text{eq, parallel}}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R} \]
Thus, the equivalent resistance for the parallel configuration is:
\[ R_{\text{eq, parallel}} = \frac{R}{3} \]

### Step 4: Calculate Power in Parallel
Using the power formula again for the parallel configuration:
\[ P_{\text{parallel}} = \frac{V^2}{R_{\text{eq, parallel}}} \]
Substituting \( R_{\text{eq, parallel}} \):
\[ P_{\text{parallel}} = \frac{V^2}{\frac{R}{3}} = \frac{3V^2}{R} \]

### Step 5: Substitute \( V^2 \) from Series into Parallel Power
Now, we substitute \( V^2 \) from our earlier calculation:
\[ P_{\text{parallel}} = \frac{3 \times 81R}{R} = 243 \, \text{W} \]

### Step 6: Calculate the Ratio of Power Dissipation
To find how many times the power dissipated in parallel is compared to series:
\[ \text{Ratio} = \frac{P_{\text{parallel}}}{P_{\text{series}}} = \frac{243}{27} = 9 \]

### Final Answer
The power dissipated when the three resistors are connected in parallel is **9 times** the power dissipated when they are connected in series.

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