ELC-07

Joule heating

P = I² R = V² / R dissipated as heat.

Reading speed

LossesJoule 1840

Governing equation

P=I^2 R=\dfrac{V^2}{R}

$I \in \left[0.01, 200\right]$

where

I: Current (A)
R: Resistance (Ω)
P: Heat power (W)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

P=I^2 R=\dfrac{V^2}{R}

$I \in \left[0.01, 200\right]$

$I = 0.01$ $P = 4\times 10^{-4}$

Along the curve

②

Domain of the variable — blue

I_{\min} = 0.01,\quad I_{\max} = 200

$I \in \left[0.01, 200\right]$

$I = 0.01$ $I = 0.01$

Along the curve

③

Response curve — green

P=I^2 R=\dfrac{V^2}{R}

$I \in \left[0.01, 200\right]$

$I = 0.01$ $P = 4\times 10^{-4}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}P}{\mathrm{d}I}

$I \in \left[0.01, 200\right]$

$I = 1.268$ $P^{\prime} = 10.14$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-07 — Joule heating) is the form associated with Joule 1840. Working symbols: $I$ , $R$ $\rightarrow$ $P$ . Joule showed that the heat released in a conductor is proportional to I² R t, the electrical work against resistance.

Purpose

Purpose: compute

P

from

I

R

in Electrical via

P=I^2 R=\dfrac{V^2}{R}

P = I² R = V² / R dissipated as heat. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

I = 5.000\,\mathrm{A}

R = 4.000\,\mathrm{Ω}

, the governing relation

P=I^2 R=\dfrac{V^2}{R}

yields

P = 100.000\,\mathrm{W}

. DC or RMS AC in a lumped resistor. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Current I5 AResistance R4 Ω

Outputs

Heat power P100.000 W

Reading speed

Watch on YouTube

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Narration of this film

DC or RMS AC in a lumped resistor.

Joule showed that the heat released in a conductor is proportional to I² R t, the electrical work against resistance.

Reading speed

Watch on YouTube