Use P = V²/R for Rated Bulbs, and Reduce Networks Step by Step
Circuit questions on bulbs, heaters and resistor networks all rest on three relations: power P = V I = I² R = V²/R, resistances that add in series and combine reciprocally in parallel, and the fact that the same heat needs the same total energy.
Electricity questions depend on identifying charge, a conducting path and the condition that controls current or magnetic effect.
Rated bulbs and power in a series circuit
Joined in series across 440 V, they carry the same current I = 440/(484 + 807) = 0.34 A, so the powers are I² R: 56 W in the 100 W bulb (less than its rating) and 94 W in the 60 W bulb (more than its rating), so it is the lower-wattage bulb that burns out. In a series circuit the larger resistance dissipates more power; in a parallel circuit, with the same voltage, the smaller resistance does.
Heating times add as rates
Heating the same water to the same temperature needs the same energy Q, so time is inversely proportional to power.
Reducing a network
Series resistances add, parallel resistances combine as 1/R = Σ 1/Rᵢ, and a symmetric network can be simplified by joining points that are at the same potential. A symmetric ladder of 2 ohm and 3 ohm resistors that reduces to 6 ohm across 18 V draws 3 A and dissipates 18 x 3 = 54 W. In a parallel split, the current divides inversely to the resistance, so a branch with a higher resistance carries less current.
in series with R0 = R/3 = 1 ohm the total is 6 ohm, so a 9 V source drives 1.5 A, V_AB = 1.5 x 5 = 7.5 V, the power in R0 is 1.5² x 1 = 2.25 W and the power in the network is 1.5² x 5 = 11.25 W.
OlympIQ exam lens
- ✓P = V I = I² R = V²/R; the rated resistance of a bulb is V²/P
- ✓Same current in series: the larger resistance dissipates more power; same voltage in parallel: the smaller resistance does
- ✓Parallel heaters: powers add, so 1/t = 1/t1 + 1/t2 for the same heat
- ✓Series: R adds; parallel: 1/R adds; join equipotential points in a symmetric network
- ✓Current divides inversely to resistance among parallel branches
Do not assume the higher-wattage bulb is the brighter or more stressed one in a series circuit. In series the same current flows, so the bulb with the larger resistance (the lower-wattage one) dissipates the most power and is the one at risk.
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