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Two exactly similar heating resistances are connected (i) in series, and (ii) in parallel, in two different circuits, one by one. If the same current is passed through both the combinations, is more heat obtained per minute when they are connected in series or when they are connected in parallel? Give reason for your answer.
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Heat Generation in Series vs. Parallel
When two identical heating resistances are connected in series and parallel, the heat generated can be compared by using Joule's law, which states that the heat produced (H) in a resistor is proportional to the square of the current (I) and the resistance (R):
H = I²Rt
Here’s how the configurations differ:
1. Series Connection
- Total Resistance: In series, the total resistance (R_total) is the sum of the individual resistances:
R_total = R1 + R2 = 2R
- Current: The same current (I) flows through both resistors.
- Heat Produced: The heat generated in series is:
H_series = I²(2R)t = 2I²Rt
This means double the heat from one resistor over time t.
2. Parallel Connection
- Total Resistance: In parallel, the total resistance (R_total) is given by:
1/R_total = 1/R1 + 1/R2 = 1/(R/2) = R/2
- Current Distribution: The total current (I) splits equally between the two resistors, so each resistor carries I/2.
- Heat Produced: The heat generated in parallel is:
H_parallel = 2 * ((I/2)²Rt) = 2 * (I²/4)Rt = (1/2)I²Rt
Here, each resistor produces half the heat of a single resistor in the same time period.
Conclusion
- Comparison:
- In series, the total heat produced is 2I²Rt.
- In parallel, the total heat produced is (1/2)I²Rt.
- Result: More heat is obtained when the resistors are connected in series than in parallel. Thus, for the same current, series connections yield greater heat generation.
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Two exactly similar heating resistances are connected (i) in series, and (ii) in parallel, in two different circuits, one by one. If the same current is passed through both the combinations, is more heat obtained per minute when they are connected in series or when they are connected in parallel? Give reason for your answer.
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