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 –

NCERT Question 4 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
Explanation: We know that,
R = 𝜌 𝑙/𝐴
Since 𝜌, l and A are same,
the resistance of two wires will be same.
Let the resistance = R
Series
We know that,
In series combination
equivalent resistance is given by
RS = R1 + R2
= R + R
= 2 R
Parallel
We know that,
In parallel combination
equivalent resistance is given by
1/𝑅_𝑝 = 1/𝑅_1 + 1/𝑅_2
1/𝑅_𝑝 = 1/𝑅 + 1/𝑅
1/𝑅_𝑝 = 2/𝑅
Rp = 𝑅/2
The potential difference in both cases in same .
Let potential difference = V
Time taken in both cases would be same = t
We know that,
Heat generated is given by
H = I2Rt
H = (𝑉/𝑅)^2Rt
H = 𝑉^2/𝑅^2 × Rt
H = 𝑉^2/𝑅 × t
We know that
V = IR
I = 𝑉/𝑅
Series
HS = 𝑉^2/𝑅_𝑆 t
HS = 𝑉^2/((2𝑅)) t
HS = (𝑉^2 𝑡)/2𝑅
Parallel
Hp = 𝑉^2/𝑅_𝑝 t
Hp = 𝑉^2/(𝑅/2) t
HS = (2 𝑉^2)/𝑅 t
Finding ratio
𝐻_𝑆/𝐻_𝑝 = (((𝑉^2 𝑡)/(2 𝑅)))/(((2𝑉^2 𝑡)/𝑅) )
𝐻_𝑆/𝐻_𝑝 = (𝑉^2 𝑡)/(2 𝑅) × 𝑅/(2 𝑉^2 𝑡)
𝐻_𝑆/𝐻_𝑝 = 1/4
∴ HS : Hp = 1 : 4
Hence, the ratio is 1 : 4
∴ Correct answer is (c)

Made by

Maninder Singh

CA Maninder Singh is a Chartered Accountant for the past 14 years and a teacher from the past 18 years. He teaches Science, Economics, Accounting and English at Teachoo

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