The lesson
1.3 Ohm’s law and power
Outcome 3 of 9 in domain 1 · Foundations, direct current
Calculate, predict heating.
Understand
Three quantities, one relation: voltage pulls, resistance holds back, current is what gets through. I = V ÷ R. The rest of this outcome is learning to read it in all three directions without thinking, and to feel it: double the voltage and current doubles; double the resistance and it halves.
It was not always obvious. In 1827, Georg Ohm teaches mathematics at a secondary school in Cologne. The batteries of his day drift while you measure, so he takes a thermocouple, two metals whose heated junction holds a steady voltage, and a magnetized needle to read the current; he borrows from Fourier, who had just written up heat flowing through a bar, the idea of a flow proportional to a difference. What comes out is a straight line. Learned Germany wants none of it: a critic calls it a web of naked fancies, a minister judges that a professor who preaches such heresies is unworthy to teach, and Ohm leaves his post. London recognizes him in 1841, Munich gives him a chair in 1852, and the telegraph engineers give his name to the unit in the 1860s. The path before the formula, here, is fifteen years of contempt for a proportionality.
Power is the second idea, and it is the one that heats, melts and burns. P = V × I, hence also P = I²R and P = V² ÷ R. The I²R form is the one to keep in mind at the bench: at equal current, it is the resistance that heats, and a wire that is too thin is a resistance. A tool pack pushing 30 A through a 5 mΩ connector leaves 4.5 W there, in a part the size of a fingernail.
Now the wire. Ten metres of AWG 16 cord make twenty metres of copper, out and back, at 13 mΩ per metre: a quarter of an ohm. At 15 A, that is 3.9 V lost and 59 W of heat spread along the cord, which gets warm. The same job in AWG 12, 5.2 mΩ per metre, loses 1.6 V and 23 W. And the same job at 48 V instead of 12 takes four times less current, hence sixteen times less heat in the same wire: that is one of the reasons for tool packs at 36, 48 and 56 V, and it holds for everything the current goes through, the windings, the cells, the contacts and the wire, which all heat as I²R. Gauge is chosen on two criteria, the heat the wire can stand and the drop the load can tolerate, and at low voltage it is almost always the drop that decides.
The classic mistake: believing that a "10 Ω" resistor is a 10 Ω resistor in every case. It also has a maximum power, often a quarter of a watt for the small ones. At 12 V, 10 Ω let 1.2 A through and dissipate 14 W: the part is dead before you have finished reading the meter. You always choose a resistor by its value and by its power, with a margin of two.
What the source lacks
Low voltage. Kuphaldt does the gauge calculation by drop, but on a 230 V, 25 A line, where the wire that does not overheat is rarely enough; he does not say that at 12 V, three percent of drop is 0.36 V and a motor feels it. The three-percent rule for low-voltage DC, and the wire the neighbour at the end of the cord will tolerate, are ours.
The connector as a resistance. A new XT60 is one or two milliohms, a worn one five or ten; at 30 A, that is the difference between warm and brown. Kuphaldt has no milliohms in his chapter 2; at the bench, they are the only ones that count.
See
The same circuit in the simulator: touch the resistor or the source to change their value, and watch the current.
The targets
- Calculate the third quantity. Three exercises in a row, without a table or a calculator for round values; it is automatic or it is not.
- Predict the power dissipated and choose the rating. For a given circuit, write the power, then the resistor you would buy, with the margin.
- Measure V and I and recover R, within 5 %. A real circuit, two measurements, one calculation, and the gap from the marked value: the board’s first proof by instrument.
- The wire that is too thin. Calculate the heating of a wire for a given current, then choose the gauge that holds, heat and drop.
- The AWG table and voltage drop. Ten metres of cable, a current, the drop in volts and what it does to the motor at the end.
- Diagnosis: the burnt resistor. On a schematic, say what it was doing, what it protected, and why it died: overvoltage, a short downstream, or a rating chosen without margin.
Misconceptions
"A thick wire carries more current."
The load decides the current, by its resistance; the wire only decides what it loses along the way, in volts and in heat. A thick wire draws nothing more, it loses less."Ohm’s law holds for everything."
It holds for a resistance whose value does not move. A lamp filament resists ten times more hot than cold; an LED has no resistance in Ohm’s sense, it has a threshold. Kuphaldt says so in "Nonlinear conduction": the law is a law of materials, and materials have conditions."A 1 W resistor dissipates 1 W."
It does, at 70 °C ambient according to its datasheet, and its surface then runs near 150 °C: it browns the board, drifts, and ages fast. The margin of two is not distrust of the datasheet, it is what you pay for a lukewarm part that lasts ten years.
The word
Ohm
Georg Simon Ohm, 1789-1854, son of a locksmith in Erlangen who had taught himself mathematics and taught it to his sons. The unit of resistance took his name in the 1860s, among British telegraph engineers, and the Paris congress confirmed it in 1881. The symbol is a Greek letter, Ω, proposed by William Preece in 1867 for a reason of the ear: omega is pronounced like the man’s name. In French, ohm, the same word; and the law, loi d’Ohm, without the possessive English kept.
Read
Kuphaldt, Lessons in Electric Circuits, volume I (DC), chapter 2: Ohm’s Law. "How voltage, current, and resistance relate", "Calculating electric power" and "Resistors" are the whole outcome; "Nonlinear conduction" is the door to the diodes of domain 3. Wire is in chapter 12, Physics of conductors and insulators: "Specific resistance" gives resistance per length and does the gauge calculation by drop, "Conductor ampacity" the one by heat.
Bridges
- Physics
- Ohm’s law is a law of materials before it is a law of circuits: resistivity, and the drift velocity of electrons, surprisingly slow, Ohm’s law.
- Mathematics
- A proportionality in three variables: knowing how to isolate it in all three directions, and reading a straight line whose slope is the resistance, Proportionality).
- History
- Georg Ohm publishes in 1827, and his colleagues despise him for fifteen years; the unit has carried his name since the 1860s, confirmed in Paris in 1881, Georg Ohm.
- Chemistry
- Copper oxide is a semiconductor: a browned contact is a resistor that was not on the schematic, Copper(II) oxide_oxide).
What it unlocks
What next?
- 1.4 Series, parallel, mixed: the same law, but several resistors at once, and the voltage divider.
- The drop over ten metres: the AWG table checked with a voltmeter, at both ends of a cord.
- Why does a contact heat up?: I²R in five milliohms.
- 1.2 The quantities: the watt, and why it is the one that burns.