Question

two conducting wires of the same material, and of equal length 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.plss answer with all correct steps​

Answer 1

Answer :-

Ratio of heat produced in series and parallel combinations would be 1 : 4

Explanation :-

Since the wires are made up of same material, have same length and equal diameters, so their resistances will be equal.

For series connection :-

Equivalent resistance (Rₛ) :-

⇒ Rₛ = R + R

⇒ Rₛ = 2R

Heat produced (H) :-

⇒ H = V²t/R

⇒ H = V²t/2R ----(1)

For parallel connection :-

Equivalent resistance (Rₚ) :-

⇒ 1/Rₚ = 1/R + 1/R

⇒ 1/Rₚ = 2/R

⇒ Rₚ = R/2

Heat produced (H') :-

⇒ H' = V²t/R

⇒ H' = V²t/(R/2)

⇒ H' = 2V²t/R ----(2)

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On dividing eq.1 by eq.2, we get :-

⇒ H/H' = V²t/2R × R/2V²t

⇒ H/H' = 1/4

⇒ H : H' = 1 : 4

Answer 2

Given:-

• Two conducting wires of the same material, and of equal length and equal diameters are first connected in series and then parallel in a circuit across the same potential difference.

To Find:-

• The ratio of heat produced in series and parallel combinations.

Formula Used:-

• ${\boxed{\bf{R_s = R+R}}}$

• ${\boxed{\bf{\dfrac{1}{R_p} = \dfrac{1}{R}+\dfrac{1}{R}}}}$

• ${\boxed{\bf{H = \dfrac{V^2t}{R}}}}$

Here,

• $R_s = Equivalent\: Resistance\:in\: Series$

• $R_p = Equivalent\: Resistance\:in\: Parallel$

• H = Heat Produced

• V = Potential Difference

• R = Resistance

• t = time

Solution:-

Equivalent Resistance in Series Combination:-

$:\implies\:R_s = R+R$

$:\implies\:R_s = 2R$

Equivalent Resistance in Parallel Combination:-

$:\implies\:\dfrac{1}{R_p} = \dfrac{1}{R}+\dfrac{1}{R}$

$:\implies\:\dfrac{1}{R_p} = \dfrac{1+1}{R}$

$:\implies\:\dfrac{1}{R_p} = \dfrac{2}{R}$

$:\implies\:R_p = \dfrac{R}{2}$

Now, Heat Produced in Series Combination:-

$\sf :\implies\:H_{(Series)} = \dfrac{V^2t}{R_s}$

$\sf :\implies\:H{(Series)} = \dfrac{V^2t}{2R}$

And, Heat Produced in Parallel Combination:-

$\sf :\implies\:H_{(Parallel)} = \dfrac{V^2t}{R_p}$

$\sf :\implies\:H_{(Parallel)} = \dfrac{V^2t}{\dfrac{R}{2}}$

$\sf :\implies\:H_{(Parallel)} = \dfrac{2V^2t}{R}$

Ratio of heat produced in series and parallel combinations:-

$\sf :\implies\: \dfrac{Heat_{(Series)}}{Heat_{(Parallel)}}$

$\sf :\implies\: \dfrac{\dfrac{V^2t}{2R}}{\dfrac{2V^2t}{R}}$

$\sf :\implies\:\dfrac{V^2t \times R}{2R \times 2V^2t}$

$\bf :\implies\:\dfrac{1}{4}$

Hence, The ratio of heat produced in series and parallel combinations is 1 : 4.

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