Question

$\huge { \red{ \bold{Question :-}}}$
Two conducting wires of the same material and of equal lengths 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 _____.

(a) 1:2

(b) 2:1

(c) 1:4

(d) 4:1

✔Need proper explained answer.
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Answer 1

Given :

Two conducting wires of equal lengths and equal diameters are first connected in series and then in parallel across same pd.

To Find :

Ratio of heat produced in series and parallel combination$.$

Solution :

In physics, long question doesn't mean that it is tough! :) In this question, meaning of 'Two conducting wires of equal lengths and equal diameters' is only that, both wires are of same resistance.

Let resistance of each wire be r.

1. Series connection :

Eq. resistance of series :

➠ R = R₁ + R₂

➠ R = r + r

➠ R = 2r

Current flow in circuit :

➠ I = V/2R .......... (I)

2. Parallel connection :

Eq. resistance of parallel :

➛ 1/R' = 1/R₁ + 1/R₂

➛ 1/R' = 1/r + 1/r

➛ 1/R' = 2/r

➛ R' = r/2

Current flow in circuit :

➛ I' = V / (r/2)

➛ I' = 2V/r .......... (II)

Heat produced in circuit is given by

• H = V × I × t

In this question, voltage of battery is same for both cases and time is also. So we can say that, heat produced in circuit is directly proportional to the current flow.

• (H ∝ I)

➝ H / H' = I / I'

➝ H / H' = (V/2r) / (2V/r)

➝ H / H' = V/2r × r/2V

➝ H : H' = 1 : 4

∴ (C) is the correct answer!

Answer 2

Given :

Two conducting wires of the same material and of equal lengths 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 would be.

Solution :

Let the resistance of two wires be R and R as they both are of equal length.

First they are connected in series, we know :

⇒ Rs = R₁ + R₂ + R₃ + ...........

⇒ Rs = R + R

⇒ Rs = 2R

Now when they are connected in parallel, so we know,

⇒ 1/Rp = 1/R₁ + 1/R₂ + ..............

⇒ 1/Rp = 1/R + 1/R

⇒ 1/Rp = (1 + 1)/R

⇒ 1/Rp = 2/R

⇒ 2Rp = R

⇒ Rp = R/2

Now, we know that :

⇒ Heat produced = V²t/R

∴ Heat produced in series combination = V²t/Rs

∴ Heat produced in parallel combination = V²t/Rp

Now their ratio :

⇒ Hs : Hp = V²t/Rs : V²t/Rp

⇒ Hs/Hp = (V²t/Rs)/(V²t/Rp)

⇒ Hs/Hp = (V²t/2R) * { (R/2) /V²t }

⇒ Hs/Hp = (R/2)/2R

⇒ Hs/Hp = R/2 * 1/2R

⇒ Hs/Hp = 1/4

⇒ Hs : Hp = 1 : 4

Therefore,

Ratio of heat produced in series and parallel combinations is 1 : 4   [Option : c]