# Two copper wires of the same material and of equal lengths and equal diameters are first connected in series and then in parallel in a circuit across a potential difference.  What would be the ratio of the heat produced in series and in parallel combination? If Rp is the total resistance of the wires in parallel connection, why there is a need to write Rp as 1/Rp while calculating the total resistance of the wire in parallel?

Two Copper wires of the same material, the same length, the same area of cross-section or diameter have the same resistance, say R.  If they are connected

a) in series,

The equivalent resistance $R_{s} = R + R = 2R$

Heat produced = $I^2 \times R_{s} \times t= I^2(2R) \times t= 2R \times I^2 \times t$

Effective or equivalent resistance doubles in series circuit

If the two copper wires are connected

b) in parallel

The equivalent resistance is $R_{p}$

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

$So R_{p} = \frac{R}{2}$

Effective or equivalent resistance halves in a parallel circuit

Heat = $I^2R_{p} \times t = I^2 \times \frac{R}{2} \times t = \frac{R}{4} \times I^2 \times t$

The ratio of heat = in series/in parallel

$\frac{2R \times I^2 \times t}{ \frac{R}{4} \times I^2 \times t}$

= 8/1 or 8:1

For parallel connection

$R_{p} = \frac{R \times R}{(R + R)} = \frac{R}{2}$

if you do not want to write

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

which is the same thing

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Updated on: 10-Oct-2022

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