Two conducting wires of the same material and of equal lengths and equal diameters are first connected in parallel and then in series in a circuit across the same potential difference. The ratio of heat produced in parallel and series combinations is
- (a)2 : 1
- (b)4 : 1
- (c)1 : 2
- (d)1 : 4
Correct — B, 4 : 1. With two equal wires (each of resistance R) across the same potential difference, the parallel combination has R = R/2 and the series combination has R = 2R. Heat produced in a fixed time is H = V^2 t / R, so the ratio is (V^2/(R/2)) : (V^2/2R) = (2V^2/R) : (V^2/2R) = 4 : 1.
- (a)2 : 1 — This treats heat as proportional to a factor of 2, but the equivalent resistances differ by a factor of 4 (2R versus R/2), and at fixed voltage heat is proportional to 1/R, giving 4 : 1.
- (c)1 : 2 — This assumes heat rises with resistance; at constant voltage heat is proportional to 1/R, so the lower-resistance parallel network makes more heat, not less.
- (d)1 : 4 — Right magnitude but inverted — the parallel combination produces four times the series heat, so the ratio is 4 : 1, not 1 : 4.
At a fixed potential difference, the heat dissipated by a resistance in a given time is H = V^2 t / R (Joule heating). Lower resistance draws more current and dissipates more heat. Connecting equal resistors in parallel lowers the total resistance to R/2, while in series it raises it to 2R, so the parallel network dissipates more heat.
The key is to use H = V^2 t / R (constant voltage), not H = I^2 R t (which would need the current). Parallel gives R/2, series gives 2R, so the heat ratio is (2R)/(R/2) = 4 : 1 in favour of the parallel combination.
- At constant voltage, heat produced H = V^2 t / R, so H is proportional to 1/R.
- Two equal resistors give R/2 in parallel and 2R in series — a 4 : 1 resistance ratio.
- Hence heat(parallel) : heat(series) = 4 : 1.
- The parallel (lower-resistance) combination always dissipates more power at the same voltage.
Because heat is proportional to 1/R at fixed voltage, the lower-resistance parallel network makes four times the heat — option (b).
- Use H = V^2 t / R at constant voltage — not H = I^2 R t, which needs the current.
- Parallel means lower resistance and MORE heat; do not invert the ratio to 1 : 4.
Compares heat or power for series versus parallel combinations at the same voltage, testing whether you choose the right power formula.
Two wires have their lengths, diameters and resistivities, all in the ratio of 1 : 2. If the resistance of the thinner wire is 10 ohms, the resistance of the thicker wire is
- (a) 10 ohms
- (b) 5 ohms
- (c) 20 ohms
- (d) 40 ohms
Answer(a) 10 ohms — with R = rho.L/A and A proportional to d^2, the two effects cancel and the resistances are equal.
UPSC prelims tested the resistance of wires from their length, diameter and resistivity (R = rho.L/A) — the same resistance reasoning behind this series/parallel heat problem.
Two resistances of 5.0 Ω and 7.0 Ω are connected in series and the combination is connected in parallel with a resistance of 36.0 Ω. The equivalent resistance of the combination of three resistors is
- (a) 24.0 Ω
- (b) 12.0 Ω
- (c) 9.0 Ω
- (d) 6.0 Ω
Answer(c) 9.0 Ω
A previous NDA GAT item computing the equivalent resistance of resistors in a series-and-parallel combination — the same circuit-combination concept.
- practice — not a real PYQ
Two identical resistors are connected first in series and then in parallel across the same battery. The ratio of total power in series to that in parallel is
- (a)4 : 1
- (b)1 : 4
- (c)2 : 1
- (d)1 : 2
Answer(b) 1 : 4 — the series combination dissipates one-fourth of the parallel power.
- practice — not a real PYQ
Household electrical appliances are connected in parallel mainly because
- (a)it saves wire
- (b)each appliance gets the full supply voltage and works independently
- (c)it increases the total resistance
- (d)it reduces the current in each appliance to zero
Answer(b) each appliance gets the full voltage and can be switched on or off independently.