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Formulas/physics/Current Electricity/Joule's Law of Heating

Joule's Law of Heating

Heat produced in a resistor carrying current I for time t. H = Pt. In calories: H (cal) = I²Rt/4.18. Basis of all resistive heating devices.
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Derivation

Derivation

In a resistor, the work done by the electric field on drifting electrons goes entirely into heat (the electrons lose their acquired drift velocity in collisions with lattice ions, increasing lattice vibration energy — heat).

Energy dissipated in time tt:

H=Pt=VItH = Pt = VIt

Using V=IRV = IR:

H=I2Rt=V2RtH = I^2Rt = \frac{V^2}{R}t H=I2Rt=VIt=V2Rt\boxed{H = I^2Rt = VIt = \frac{V^2}{R}t}

Joule's laws (stated)

  1. H∝I2H \propto I^2 at constant RR and tt
  2. H∝RH \propto R at constant II and tt
  3. H∝tH \propto t at constant II and RR

Units and conversion

[H]=J[H] = \text{J}. In calories: H (cal)=H (J)/4.18H\,(\text{cal}) = H\,(\text{J})/4.18

The mechanical equivalent of heat: 1 cal=4.18 J1\,\text{cal} = 4.18\,\text{J}.

Applications

Joule heating is the operating principle of incandescent bulbs, electric heaters, fuses, and electric irons. In all cases, the desired effect (light, heat, melting) relies on H=I2RtH = I^2Rt.

Note
Joule heating is always wasteful in transmission lines ($P_{loss} = I^2R$). High-voltage transmission reduces $I$ for the same power ($P = VI$), dramatically cutting $I^2R$ losses. This is the primary reason for high-voltage AC transmission.