Two resistances, 10 Ω and 20 Ω, are connected in series circuit A and then in parallel in circuit B with a battery of 24 V.
Choose the correct statements.
(A) Power dissipation is greater in circuit B.
(B) Power dissipation is greater in circuit A.
(C) Current is larger in circuit B.
(D) Ratio of power dissipation in circuit B to power dissipation in circuit A is 9:2
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
With the same battery voltage V = 24 V, the power drawn is \( P = V^{2}/R \), so the circuit with the smaller equivalent resistance draws more current and dissipates more power.
\( R_A = 10\,\Omega + 20\,\Omega = 30\,\Omega \)
\( \displaystyle R_B = \frac{10 \times 20}{10 + 20}\,\Omega = \frac{20}{3}\,\Omega \)
\( \displaystyle P_A = \frac{(24\,\mathrm{V})^{2}}{30\,\Omega} = 19.2\,\mathrm{W} \)
\( \displaystyle P_B = \frac{(24\,\mathrm{V})^{2} \times 3}{20\,\Omega} = 86.4\,\mathrm{W} \)
\( I_A = 24/30 = 0.8\,\mathrm{A} \)
\( I_B = 24 \times 3/20 = 3.6\,\mathrm{A} \)
(A) Power dissipation is greater in circuit B. — Correct. 86.4 W is greater than 19.2 W.
(B) Power dissipation is greater in circuit A. — Incorrect. The series circuit has the larger resistance, so it dissipates less power.
(C) Current is larger in circuit B. — Correct. 3.6 A is drawn from the battery in circuit B against 0.8 A in circuit A (even each resistor in B, with 2.4 A and 1.2 A, carries more than 0.8 A).
(D) Ratio of power dissipation in circuit B to power dissipation in circuit A is 9:2 — Correct.
\( \displaystyle \frac{P_B}{P_A} = \frac{86.4\,\mathrm{W}}{19.2\,\mathrm{W}} = \frac{9}{2} \)
Shortcut: At the same voltage \( P \propto 1/R \), so \( P_B/P_A = R_A/R_B = 30 \div (20/3) = 9/2 \) directly.