Bright or long-lasting: choosing a lamp circuit
A learner builds lamp circuits with a 6.0 V battery pack and two identical bulbs marked '6.0 V, 0.30 A'. In arrangement 1 the bulbs are connected one after the other and the ammeter reads 0.15 A; both glow dimly. In arrangement 2 each bulb is connected directly across the battery and the ammeter in the main branch reads 0.60 A; both glow brightly. After three minutes in arrangement 2 the battery pack feels warm. The learner has no spare bulbs and wants light that is as bright as possible but lasts as long as possible on one battery.
1. Use the bulb rating and the two ammeter readings to show that the circuit behaviour is consistent.
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From the rating, R = V/I = 6.0 V / 0.30 A = 20 Ω per bulb.
Arrangement 1 (series): total resistance = 20 + 20 = 40 Ω, so I = V/R = 6.0/40 = 0.15 A. This matches the ammeter. Each bulb now has only 3.0 V across it, so each glows dimly.
Arrangement 2 (parallel): each bulb has the full 6.0 V across it, so each takes 0.30 A, and the main branch carries 0.30 + 0.30 = 0.60 A. This matches the second reading.
Both readings are exactly what the simple model predicts, so the learner can trust the circuit and use the model to predict other arrangements. One caveat: a filament's resistance changes with temperature, so 20 Ω is a working value rather than a constant.
2. Explain why the battery pack warms up in arrangement 2, and compare how long each arrangement will run on one battery.
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Power delivered: series P = VI = 6.0 × 0.15 = 0.90 W; parallel P = 6.0 × 0.60 = 3.6 W, four times as much.
A real battery has some internal resistance, and heating inside it is I²r. At four times the current it produces sixteen times the internal heating, which is easily enough to feel warm after three minutes.
Because the parallel circuit uses chemical energy four times as fast, it should run for about a quarter as long: if the series arrangement lasts four hours, the parallel one lasts roughly one. So 'brighter' and 'lasts longer' pull in opposite directions here - the learner cannot have both with a single battery, and must decide which matters more.
3. Recommend an arrangement and justify it against the learner's stated aim.
Read the worked answer
The aim contains a conflict, so the first step is to state a priority.
If the light must last as long as possible - a power cut at night - use arrangement 1: 0.90 W, about 0.15 A, both bulbs lit, though dim.
If brightness matters more, use arrangement 2 and accept short life.
The best compromise with the equipment available is a single bulb connected directly across the battery. It takes 0.30 A and is fully bright, drawing half the parallel current, so it lasts twice as long as arrangement 2 while giving more light than either bulb in series. It also leaves one bulb unused as a spare, which is useful because an over-run bulb is the part most likely to fail.
The recommendation must name the criterion used and the trade-off accepted, since there is no single correct answer.