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 = ๐ ๐/๐ด
Since ๐, 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
1/๐ _๐ = 1/๐ _1 + 1/๐ _2
1/๐ _๐ = 1/๐ + 1/๐
1/๐ _๐ = 2/๐
Rp = ๐ /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 = (๐/๐ )^2Rt
H = ๐^2/๐ ^2 ร Rt
H = ๐^2/๐ ร t
We know that
V = IR
I = ๐/๐
Series
HS = ๐^2/๐ _๐ t
HS = ๐^2/((2๐ )) t
HS = (๐^2 ๐ก)/2๐
Parallel
Hp = ๐^2/๐ _๐ t
Hp = ๐^2/(๐ /2) t
HS = (2 ๐^2)/๐ t
Finding ratio
๐ป_๐/๐ป_๐ = (((๐^2 ๐ก)/(2 ๐ )))/(((2๐^2 ๐ก)/๐ ) )
๐ป_๐/๐ป_๐ = (๐^2 ๐ก)/(2 ๐ ) ร ๐ /(2 ๐^2 ๐ก)
๐ป_๐/๐ป_๐ = 1/4
โด HS : Hp = 1 : 4
Hence, the ratio is 1 : 4
โด Correct answer is (c)

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.