Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be :(a) 1 : 2(b) 2 : 1(c) 1 : 4(d) 4 : 1
Question
Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then parallel in a circuit across the same potential difference. The ratio of heat produced in series and parallel combinations would be :(a) 1 : 2(b) 2 : 1(c) 1 : 4(d) 4 : 1
Solution
The heat produced in a resistor (or a wire, in this case) is given by Joule's law, which states that the heat produced is directly proportional to the square of the current (I^2) and the resistance (R), and the time (t) for which the current flows. This can be written as H = I^2Rt.
When the wires are connected in series, the total resistance is the sum of the individual resistances. If each wire has a resistance of R, the total resistance in series is 2R.
When the wires are connected in parallel, the total resistance is less than the resistance of the smallest resistor. For two resistors of equal resistance, the total resistance in parallel is R/2.
The potential difference (V) across the wires is the same in both cases. Ohm's law states that the current is the potential difference divided by the resistance, or I = V/R.
Therefore, the heat produced in the series case is H_series = (V/(2R))^2 * 2R * t = V^2*t/(2R).
The heat produced in the parallel case is H_parallel = (V/(R/2))^2 * (R/2) * t = 2V^2t/R.
The ratio of the heat produced in the series case to the parallel case is therefore H_series/H_parallel = (V^2t/(2R)) / (2V^2*t/R) = 1/4.
So, the correct answer is (c) 1 : 4.
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