# Two conducting wires of the same material

Question.
Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then in parallel in an electric circuit. The ratio of the heat produced in series and parallel combinations would be

(1) 1 : 2

(2) 2 : 1

(3) 1 : 4

(4) 4 : 1

solution:

Since both the wires are made of the same material and have equal lengths and equal diameters,

they have the same resistance. Let it be R.

When connected in series, their equivalent resistance is given by

$\mathrm{R}_{\mathrm{s}}=\mathrm{R}+\mathrm{R}=2 \mathrm{R}$

When connected in parallel, their equivalent resistance is given by

$\frac{1}{R_{p}}=\frac{1}{R}+\frac{1}{R}=\frac{2}{R}$ or $R_{p}=\frac{R}{2}$

Further, electrical power is given by $P=\frac{V^{2}}{R}$

Power (or heat produced) in series, $P_{s}=\frac{V^{2}}{R_{s}}$

Power (or heat produced) in parallel, $P_{p}=\frac{V^{2}}{R_{p}}$

Thus, $\frac{P_{s}}{P_{p}}=\frac{V^{2} / R_{s}}{V^{2} / R_{p}}=\frac{R_{p}}{R_{s}}=\frac{R / 2}{2 R}=\frac{1}{4}$

or $P_{s}: P_{p}=1: 4$

Thus, (3) is the correct answer.