Two conducting 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 the same potential difference. The ratio of heat produced in series and parallel combinations would be :

Question

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Answer 1

Since the two wires have same material, length and diameter, they have the same resistance. Let the resistance of each wire be R.
Let applied potential difference be $V$ .

$R_{s e r i e s} = R_{1} + R_{2}$
$= R + R = 2 R$
$P_{s e r i e s} = V^{2} / R_{s e r i e s}$
$= V^{2} / (2 R) . . . . . . . . . . (i)$

$\frac{1}{R _{p a r a l l e l}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}$
$R_{p a r a l l e l} = R / 2$

$P_{p a r a l l e l} = V^{2} / R_{p a r a l l e l}$
$= 2 V^{2} / R . . . . . . . . . . . . (i i)$

Dividing (ii) from (i),
$\frac{P _{s e r i e s}}{P _{p a r a l l e l}} = 1 / 4$

Answer 2

Since the two wires have same material, length and diameter, they have the same resistance. Let the resistance of each wire be R.

Let applied potential difference be

V

.

R_{s e r i e s} = R_{1} + R_{2}

= R + R = 2 R

P_{s e r i e s} = V^{2} / R_{s e r i e s}

= V^{2} / (2 R) . . . . . . . . . . (i)

\frac{1}{R _{p a r a l l e l}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}

R_{p a r a l l e l} = R / 2

P_{p a r a l l e l} = V^{2} / R_{p a r a l l e l}

= 2 V^{2} / R . . . . . . . . . . . . (i i)

Dividing (ii) from (i),

\frac{P _{s e r i e s}}{P _{p a r a l l e l}} = 1 / 4