Adding Lamps in Parallel: Dimmer Lamps but Brighter Room
A student is given a 6.0 V battery of internal resistance 1.5 Ω and two identical torch lamps marked '6 V, 3 W'. With one lamp connected the lamp is bright; with both lamps in parallel across the battery each lamp is noticeably dimmer than before. He then adds a third identical lamp in parallel, expecting the total light to fall, and is surprised that the battery becomes warm and that the three lamps together give more total light than the two, although each single lamp is dimmer still. For the calculation assume each filament keeps its rated resistance, and ignore the resistance of the connecting wires and the ammeter.
He repeats the test with a fourth and then a fifth lamp and finds that beyond a certain number the total light from the lamps stops increasing, even though the battery is hotter than ever.
1. For the two-lamp circuit calculate the resistance of one lamp, the current drawn from the battery, the terminal potential difference and the power dissipated in each lamp, and compare with the 3 W rating.
Read the worked answer
Rated resistance: R = V²/P = 6.0²/3.0 = 12 Ω for each lamp.
Two in parallel: R_ext = 12/2 = 6.0 Ω; total circuit resistance = 6.0 + 1.5 = 7.5 Ω.
Current: I = E/(R_ext + r) = 6.0/7.5 = 0.80 A.
Terminal p.d.: V = E − Ir = 6.0 − 0.80 × 1.5 = 4.8 V, which checks against V = I R_ext = 0.80 × 6.0 = 4.8 V.
Each lamp: I_lamp = 4.8/12 = 0.40 A, so P = I²R = 0.40² × 12 = 1.92 W. That is only 64 % of the 3 W rating - clearly dimmer, as observed.
Power balance: external power 4.8 × 0.80 = 3.84 W; internal dissipation 0.80² × 1.5 = 0.96 W; sum 4.80 W = E I ✓.
2. Repeat for three lamps. Then resolve the paradox: how can each lamp be dimmer while the three together give more light? Why does the total stop increasing when enough lamps are added?
Read the worked answer
Three in parallel: R_ext = 12/3 = 4.0 Ω; total = 5.5 Ω. I = 6.0/5.5 = 1.09 A; V = 6.0 − 1.09 × 1.5 = 4.36 V. Each lamp: 4.36/12 = 0.364 A, P = 0.364² × 12 = 1.59 W (53 % of rating). Total light power from the lamps: 3 × 1.59 = 4.76 W, up from 3.84 W with two. Internal loss: 1.09² × 1.5 = 1.79 W, nearly double the 0.96 W - the battery heats more.
Resolution. Each lamp's brightness depends on its own p.d., and that falls because the larger current increases the 'lost volts' Ir inside the battery. The total power delivered to the lamps depends on the external resistance: adding a parallel branch lowers R_ext, raises I and raises the external power P_ext = EI − I²r, as long as R_ext > r.
The maximum. dP_ext/dR_ext = 0 when R_ext = r = 1.5 Ω, i.e. eight 12 Ω lamps in parallel. Up to that point each extra lamp adds total light while dimming the others; beyond it, extra lamps really do reduce the total external power, which is why the light stops growing and the battery simply gets hotter (all the extra energy goes into I²r).
3. Explain why the constant-resistance assumption is unsafe, and what it does to (a) the predicted lamp currents and (b) the behaviour at the instant of switching on.
Read the worked answer
A tungsten filament's resistance rises steeply with temperature - roughly tenfold between room temperature and white heat - because the hotter lattice scatters conduction electrons more strongly. The '12 Ω' is the hot, rated value, not a constant.
(a) Predicted currents. When the terminal p.d. falls to 4.36 V the filament runs cooler than at its rating, so its actual resistance is somewhat below 12 Ω. Each lamp therefore takes a little more current than 0.364 A and is a little brighter than the constant-R model says. The model is a worst-case estimate for dimming, and the real total external power is slightly higher than 4.76 W.
(b) Switch-on. At the instant of switching on the filament is cold and its resistance may be only 1–2 Ω, so the surge current can be many times the steady value; the terminal p.d. collapses momentarily and the lamps often fail at that instant. The model also ignores the rise of the battery's internal resistance as it warms, which increases the lost volts and compounds the effect on a long run.