Question

Two resistors \(R\) and \(2R\) are connected in parallel in an electric circuit. The thermal energies developed in \(R\) and \(2R\) are in the ratio:

• A. \(1 : 2\)
• B. \(2 : 1\)
• C. \(1 : 4\)
• D. \(4 : 1\)

Hint:

Parallel means equal voltage, so use \(H = V^2t/R\). Heat is inversely proportional to resistance.

Answer 1

The Correct Option is B

Step 1: Concept. In a parallel connection, both resistors have the SAME potential difference \(V\) across them. The thermal (heat) energy developed in time \(t\) is \(H = \dfrac{V^2}{R}\,t\), so at fixed \(V\) and \(t\), heat is inversely proportional to resistance.
Step 2: Write the heat in each resistor. For \(R\): \(H_R = \dfrac{V^2}{R}\,t\). For \(2R\): \(H_{2R} = \dfrac{V^2}{2R}\,t\).
Step 3: Take the ratio. \[\frac{H_R}{H_{2R}} = \frac{V^2/R}{V^2/(2R)} = \frac{2R}{R} = 2.\]
Step 4: So \(H_R : H_{2R} = 2 : 1\). This is option (ii).
Why other options are wrong: Students often use \(H = I^2Rt\) with the same current, which applies to SERIES, not parallel. In parallel the voltage is common, so the correct formula is \(V^2/R\), giving \(2:1\), not \(1:2\).
\[\boxed{H_R : H_{2R} = 2 : 1}\]