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Joule's law: $H = I^2Rt$ — heat produced in a resistor is proportional to the square of the current, resistance, and time. Equivalently: $H = VIt = V^2t/R$. This irreversible conversion of electrical energy to heat is the basis for electric heaters, fuses, and incandescent bulbs.

Power rating: a "100 W, 220 V" bulb has rated resistance $R = V^2/P = 220^2/100 = 484$ ohm. It consumes 100 W only at 220 V. At different voltages: $P = V^2/R$ (the resistance is approximately constant for the short duration of the problem).

Series vs. parallel behavior of rated bulbs is a classic JEE trap: (1) In series (same current): power $\propto R$. The higher-wattage bulb (lower $R$) glows dimmer. (2) In parallel (same voltage): each operates at its rated power. The higher-wattage bulb is brighter.

Electrical energy: $E = Pt = VIt$. Commercial unit: kilowatt-hour (kWh) $= 3.6 \times 10^6$ J. Fuse wires have low melting points and are rated for maximum safe current: $I_{\text{fuse}} = \sqrt{P_{\text{max}}/R_{\text{fuse}}}$.