What is Resistance?

Example 58.3 (Resistances around you)

A metre of thick copper wire: hundredths of an ohm — the level road. A glowing flashlight bulb: some ohms. A small electric motor: a few ohms. The kettle’s heating coil: about $25\,\Omega$. Dry human skin, hand to hand: tens of thousands of ohms — wet, catastrophically less: the old damp-hands rule is a resistance statement. An open switch: resistance beyond all measuring — the infinite obstacle.

Example 58.5 (A resistor’s confession)

A session’s table, for one resistor:

Double the push, double the march; triple, triple: $U$ and $I$ are proportional, and every ratio $U/I$ returns the same number: $15$. That constant ratio is no accident — it is the resistor’s resistance, $R = 15\,\Omega$, showing itself in every row.

Example 58.9 (Old mysteries on one invoice)

The unequal voltage split of the laws chapter: series lamps share one $I$, so by $U = R I$ the bigger $R$ takes the bigger $U$ — the harder worker’s larger drop, now computable. The throttled branch of the parallel chapter: the long thin wire added resistance, and $I = U/R$ shrank that branch’s share. The short circuit: a bypass of nearly zero $R$, so $I = U/R$ explodes — the stampede was always division by almost-zero. Four years of qualitative circuit lore, one law’s invoice.

Example 12.9 (Equivalent resistance)

$R_1 = 100\,\Omega$ and $R_2 = 150\,\Omega$: in series, $250\,\Omega$; in parallel, $\frac{100 \times 150}{250} = 60\,\Omega$ — less than either: the current gets a second road.

Example 12.12 (Reading a nameplate)

A kettle marked “$230\,\mathrm{V}$ – $2200\,\mathrm{W}$” draws $I = P/U = 2200/230 = 9.6\,\mathrm{A}$ and has resistance $R = U^2/P = 230^2/2200 = 24\,\Omega$. Running $150\,\mathrm{s}$ it uses $E = 2200 \times 150 = 3.3 \times 10^{5}\,\mathrm{J} \approx 0.09\,\mathrm{kW}\,\mathrm{h}$ — about two cents.

Example 12.15 (Why pylons run high)

A village needs $P = 100\,\mathrm{kW}$ through a line of resistance $R_\ell = 5.0\,\Omega$. Delivered at $U = 500\,\mathrm{V}$: $I = P/U = 200\,\mathrm{A}$ and the line wastes $R_\ell I^2 = 200\,\mathrm{kW}$ — twice the delivery: absurd. At $U = 20\,\mathrm{kV}$: $I = 5.0\,\mathrm{A}$ and $R_\ell I^2 = 125\,\mathrm{W}$. Forty times less current, $1600$ times less loss.