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: Set up currents. In parallel the common voltage is \(V\). By Ohm's law the branch currents are \(I_R = V/R\) and \(I_{2R} = V/(2R)\), so \(I_R = 2\,I_{2R}\).
Step 2: Use the power form of heating. Power dissipated is \(P = I^2 R\), and heat \(= P\,t\) for equal time \(t\).
Step 3: Compute each power. \(P_R = I_R^2\,R = \left(\dfrac{V}{R}\right)^2 R = \dfrac{V^2}{R}\) and \(P_{2R} = \left(\dfrac{V}{2R}\right)^2 (2R) = \dfrac{V^2}{2R}\).
Step 4: Divide. \(\dfrac{P_R}{P_{2R}} = \dfrac{V^2/R}{V^2/2R} = 2\). Since the time is common, the heat ratio is the same, \(2:1\), which is option (ii).
\[\boxed{P_R : P_{2R} = 2 : 1}\]