Question

Three equal resistances are first connected in series and afterwards connected in parallel to the same supply voltage. The ratio of the power consumed in each resistor in the series and parallel combination will respectively be-
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Answer 1

The ratio of power consumed in each resistor in the series and parallel combination is 1 : 9

Explanation:

Let the voltage applied be V

Let the equal resistances be R

We know that when resistances are connected in series

The equivalent resistance is given by the sum of all the resistances

Therefore,

In series connection, the equivalent resistance is

$R_{eq}=R+R+R=3R$

Power is given by

$P=\frac{V^2}{R_{eq}}$

$\implies P=\frac{V^2}{3R}$

Also when the resistances are connected in parallel, the reciprocal of equivalent resistance is given by the sum of the reciprocal of all the resistances

Therefore,

In parallel connection, the equivalent resistance is

$\frac{1}{R_{eq}'}=\frac{1}{R}+\frac{1}{R}+\frac{1}{R}$

$\implies \frac{1}{R_{eq}'}=\frac{3}{R}$

$\implies R_{eq}'=\frac{R}{3}$

Power is given by

$P'=\frac{V^2}{R_{eq}'}$

$\implies P'=\frac{V^2}{R/3}$

$\implies P'=\frac{3V^2}{R}$

Now

$\frac{P}{P'}=\frac{V^2/3R}{3V^2/R}$

$\implies \frac{P}{P'}=\frac{V^2}{3R}\times\frac{R}{3V^2}$

$\implies \frac{P}{P'}=\frac{1}{9}$

or, $P:P'=1:9$

Hope this answer is helpful.

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Answer 2

ratio of the power consumed in each resistor in the series and parallel combination  is  1:9

Explanation:

Let say three resistances are R  

Total resistance in Series = R + R + R = 3R

Current in Each Resistance =  V/3R     ( current is same in series)

Power in each resistance = (V/3R)² * R  = V²/9R

Resistance connected in Parallel

Current in Each Resistance = V/R    ( as Voltage is same in parallel)

Power in each resistance   = (V/R)²R = V²/R

ratio of the power consumed in each resistor in the series and parallel combination  =  (V²/9R  )/(V²/R)   =  1/9

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