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

Answer 1

Answer: (C)

1:4

Answer 2

Heat produced in the circuit is inversely proportional to the resistance $R$. Let $R_S$ and $R_P$ be the equivalent resistances of the wires if connected in series and parallel respectively.
Let $R$ be the resistance of each wire. If the resistors are connected in parallel, the net resistance is given by
$\frac {1}{𝑅_𝑃} = \frac 1𝑅+ \frac 1𝑅$

⟹ $\frac {1}{R_𝑃}= \frac 2𝑅$

⟹ $𝑅_𝑃=\frac {𝑅}{2}$
If the resistors are connected in series, the net resistance is given by
$𝑅_𝑠=𝑅+𝑅$
$R_S=2𝑅$
Hence, for same potential difference V, the ratio of heat produced in the circuit is given by
$\frac {𝐻_𝑆}{𝐻_𝑃}= \frac {\frac {𝑉^2}{𝑅_𝑠}𝑡}{\frac {𝑉^2}{𝑅_𝑃}𝑡}$

$\frac {𝐻_𝑆}{𝐻_𝑃}= \frac {𝑅_𝑃}{𝑅_𝑠}$

$\frac {𝐻_𝑆}{𝐻_𝑃} = \frac {\frac 𝑅2}{2𝑅}$

$\frac {𝐻_𝑆}{𝐻_𝑃}=\frac 14$

${𝐻_𝑆}:{𝐻_𝑃}=1:4$
Therefore, the ratio of heat produced in series and parallel combinations is 1:4.

Hence, the correct option is (C): $1:4$