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

The Correct Option is C

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): \(1:4\)