Three equal resistors connected in series across a source of e.m.f. together dissipate 10 W of power. What should be the power dissipated if the same resistors are connected in parallel across the same source of e.m.f.

Options

A. 9 W
B. 90 W
C. 10 W
D. 100 W

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the series connection of resistors
When three equal resistors (let's denote each as R) are connected in series, the total or equivalent resistance (R_eq) is given by:
\[ R_{eq} = R_1 + R_2 + R_3 = R + R + R = 3R \]

### Step 2: Relate power to resistance in series
The power dissipated (P) in a circuit can be expressed using the formula:
\[ P = \frac{V^2}{R_{eq}} \]
Given that the power dissipated in the series connection is 10 W, we can write:
\[ 10 = \frac{V^2}{3R} \]

### Step 3: Rearranging the equation to find V²/R
From the equation above, we can rearrange it to find:
\[ V^2 = 10 \times 3R = 30R \]
Thus, we have:
\[ \frac{V^2}{R} = 30 \]

### Step 4: Understand the parallel connection of resistors
When the same three resistors are connected in parallel, the equivalent resistance (R_eq_parallel) is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R} \]
Thus, the equivalent resistance for the parallel connection is:
\[ R_{eq} = \frac{R}{3} \]

### Step 5: Calculate power in the parallel connection
Using the power formula again for the parallel connection:
\[ P_{parallel} = \frac{V^2}{R_{eq}} \]
Substituting \( R_{eq} = \frac{R}{3} \):
\[ P_{parallel} = \frac{V^2}{\frac{R}{3}} = 3 \frac{V^2}{R} \]

### Step 6: Substitute the value of V²/R
From Step 3, we know that \( \frac{V^2}{R} = 30 \). Therefore:
\[ P_{parallel} = 3 \times 30 = 90 \text{ W} \]

### Conclusion
The power dissipated when the resistors are connected in parallel across the same source of e.m.f. is:
\[ \boxed{90 \text{ W}} \]

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