---
title: "Resistance and Ohm’s Law"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 58
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/58-resistance-and-ohms-law
---

# Chapter 58 — Resistance and Ohm’s Law

One question has trailed the whole electrical story: some components let the march rush, others throttle it to a trickle — lamps split [voltages](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) unequally, thin wires warm, thick ones stay cool. The opposition itself now gets a name, a [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring), and — crowning four years of circuits — the single most useful law in this book: three letters that tie the push, the march and the opposition into one line of algebra.

## 58.1 Resistance

**Definition 58.1 (Resistance).**

The *resistance* $R$ of a component [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) how strongly it opposes the [electric current](https://one-course.com/books/physics/1/en/chapter/47-circuits-loops-series-parallel#def-g7-circuits-series-parallel-current): at a given [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage), the greater the resistance, the weaker the march it lets through. Its [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) is the *ohm* (symbol $\Omega$, the Greek capital omega), honoring the schoolmaster-physicist who found this chapter’s law with homemade wires and heroic patience.

**Definition 58.2 (Resistor).**

A *resistor* is a component manufactured to have a chosen, steady [resistance](#def-g8-ohms-law-resistance) — a calibrated obstacle, sold from fractions of an [ohm](#def-g8-ohms-law-resistance) to millions. Circuits use resistors to set currents on purpose: taming the march that would burn a delicate part, dimming, dividing, adjusting. Their striped color bands spell out their value; their symbol is a plain rectangle.

**Example 58.3 (Resistances around you).**

A [metre](https://one-course.com/books/physics/1/en/chapter/7-measuring-length#def-g2-measuring-length-units) of thick copper wire: hundredths of an [ohm](#def-g8-ohms-law-resistance) — the level road. A glowing flashlight [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb): some [ohms](#def-g8-ohms-law-resistance). A small electric [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor): a few [ohms](#def-g8-ohms-law-resistance). The kettle’s heating coil: about $25\,\Omega$. Dry human skin, hand to hand: tens of thousands of [ohms](#def-g8-ohms-law-resistance) — wet, catastrophically less: the old damp-hands rule is a [resistance](#def-g8-ohms-law-resistance) statement. An open [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch): [resistance](#def-g8-ohms-law-resistance) beyond all measuring — the infinite obstacle.

## 58.2 The experiment

**Method 58.4 (Measuring a component’s UUU–III portrait).**

One [resistor](#def-g8-ohms-law-resistor), one adjustable supply (or a growing queue of cells), both meters:

1. wire supply, [ammeter](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-ammeter) and [resistor](#def-g8-ohms-law-resistor) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) ; clip the [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) across the [resistor](#def-g8-ohms-law-resistor) ;
2. set the smallest [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) ; record the pair ( $U$ , $I$ );
3. raise the [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) step by step — one cell, two, three — recording a pair each time;
4. plot the points, $I$ across, $U$ up, and look.

**Example 58.5 (A resistor’s confession).**

A session’s table, for one [resistor](#def-g8-ohms-law-resistor):

| $U$ ($\mathrm{V}$) | $1.5$ | $3.0$ | $4.5$ | $6.0$ |
| --- | --- | --- | --- | --- |
| $I$ ($\mathrm{A}$) | $0.10$ | $0.20$ | $0.30$ | $0.40$ |

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](#def-g8-ohms-law-resistor)’s [resistance](#def-g8-ohms-law-resistance), $R = 15\,\Omega$, showing itself in every row.

**Proposition 58.6 (Ohm’s law).**

For a [resistor](#def-g8-ohms-law-resistor) at steady [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature), the [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) across it and the current through it are proportional, and the constant of proportionality is its [resistance](#def-g8-ohms-law-resistance):

$$
U = R \times I ,
$$

with $U$ in [volts](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage), $I$ in [amperes](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity), $R$ in [ohms](#def-g8-ohms-law-resistance). Rearranged to taste: $I = U/R$ (the current a push drives through an obstacle) and $R = U/I$ (the obstacle, unmasked by one honest pair of readings).

![The resistor’s U–I portrait: measured points on a straight line through the origin — proportionality drawn. The steeper the line, the greater the resistance.](https://one-course.com/images/onecourse/chapters/physics-1/g8-ohms-law/fig-91cf0f35d35f.svg)

*The [resistor](#def-g8-ohms-law-resistor)’s $U$–$I$ portrait: measured points on a straight line through the origin — proportionality drawn. The steeper the line, the greater the [resistance](#def-g8-ohms-law-resistance).*

**Example 58.7 (The law computes everything).**

Once $R$ is known, the law answers in every direction. The kettle coil, $25\,\Omega$ on the $230\,\mathrm{V}$ mains: $I = U/R = 230 \div 25 = 9.2\,\mathrm{A}$ — the [ten-ampere](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity) appliance explained. A [resistor](#def-g8-ohms-law-resistor) must limit a delicate lamp’s current to $0.02\,\mathrm{A}$ from $4.5\,\mathrm{V}$: $R = U/I = 4.5
\div 0.02 = 225\,\Omega$ — pick the next standard value. Through $60\,\Omega$ flows $0.05\,\mathrm{A}$: the [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) across it is $U = 60 \times 0.05 = 3\,\mathrm{V}$. One law, three recipes, the whole workshop.

**Remark 58.8 (The triangle crutch).**

Some learners pencil $U$ over $R$ and $I$ in a little triangle, covering the wanted letter to read off the recipe. Harmless as a crutch — but your algebra now rearranges $U = R I$ honestly in one line, and the honest way travels to every formula you will ever meet, triangle or none. Lean on the crutch if you must this year; plan to walk.

## 58.3 What resistance explains

**Example 58.9 (Old mysteries on one invoice).**

The unequal [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-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](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) of the parallel chapter: the long thin wire added [resistance](#def-g8-ohms-law-resistance), and $I = U/R$ shrank that [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s share. The [short circuit](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-device): 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 58.10 (Wires, by the metre).**

A wire’s [resistance](#def-g8-ohms-law-resistance) grows with its length and shrinks with its thickness — a long thin wire is a narrow mountain road, a short thick one a motorway. Hence the electrician’s fat cables for hungry appliances, the heating coil’s deliberately long thin resistive alloy — and the reason the same law that [lights](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) a lamp warms a toaster: [resistance](#def-g8-ohms-law-resistance) is where the march spends its push, and spent push, as the [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) chapter will confirm next year, becomes warmth.

**Remark 58.11 (Where the law’s writ ends).**

Ohm’s law is a [resistor](#def-g8-ohms-law-resistor)’s law, not a universal right. Measure a *filament lamp’s* portrait and the points bend away from the straight line: the filament [heats](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) as the current grows, and hot metal resists more — $R$ refuses to stay put. Diodes, [motors](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor) and living skin bend their portraits too, each in its own way. The straight line through the origin is the [resistor](#def-g8-ohms-law-resistor)’s signature, not matter’s — always ask a component for its portrait before trusting it with the law.

## 58.4 Exercises

**Exercise 58.1 ★.**

What does [resistance](#def-g8-ohms-law-resistance) [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring)? Give its [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) and symbol, and write Ohm’s law with its three rearrangements.

**Solution of Exercise 58.1.**

How strongly a component opposes the current; [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) the [ohm](#def-g8-ohms-law-resistance), symbol $\Omega$. $U = R I$; $I = U/R$; $R = U/I$.

**Exercise 58.2 ★.**

Complete each pair for a $30\,\Omega$ [resistor](#def-g8-ohms-law-resistor): $U =
6\,\mathrm{V}$, $I = {?}$; $I = 0.5\,\mathrm{A}$, $U = {?}$.

**Solution of Exercise 58.2.**

$I = 6 \div 30 = 0.2\,\mathrm{A}$; $U = 30 \times 0.5 =
15\,\mathrm{V}$.

**Exercise 58.3 ★.**

A component shows $U = 4.5\,\mathrm{V}$ at $I = 0.09\,\mathrm{A}$. Its [resistance](#def-g8-ohms-law-resistance)?

**Solution of Exercise 58.3.**

$R = 4.5 \div 0.09 = 50\,\Omega$.

**Exercise 58.4 ★.**

In the portrait experiment, what shape do a true [resistor](#def-g8-ohms-law-resistor)’s points draw, and what does a steeper line mean?

**Solution of Exercise 58.4.**

A straight line through the origin — proportionality. A steeper line means more [volts](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) needed per [ampere](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity): greater [resistance](#def-g8-ohms-law-resistance).

**Exercise 58.5 ★.**

From the chapter’s table, verify $R$ from the second and fourth rows. Why is getting the same number both times the whole point?

**Solution of Exercise 58.5.**

$3.0 \div 0.20 = 15$; $6.0 \div 0.40 = 15$. The sameness of the ratio across rows *is* the law: one constant $R$ serving every reading is what makes “the [resistance](#def-g8-ohms-law-resistance)” a property of the component.

**Exercise 58.6 ★.**

The kettle coil is $25\,\Omega$; a second kettle’s is $35\,\Omega$. On the same mains, which draws the stronger current — and how much does each draw (one decimal)?

**Solution of Exercise 58.6.**

The smaller [resistance](#def-g8-ohms-law-resistance) drinks more: $230 \div 25 =
9.2\,\mathrm{A}$ against $230 \div 35 \approx 6.6\,\mathrm{A}$.

**Exercise 58.7 ★.**

Explain with $U = R I$ why, of two series lamps sharing one current, the [higher-resistance](#def-g8-ohms-law-resistance) lamp takes the larger [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage).

**Solution of Exercise 58.7.**

Series lamps share one $I$; each lamp’s [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) is $U = R I$ with the same $I$ — so the larger $R$ multiplies into the larger $U$: the stronger opposer takes the bigger share.

**Exercise 58.8 ★.**

Translate into resistance-language: the [short circuit](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-device)’s stampede; the open [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch)’s total blockade.

**Solution of Exercise 58.8.**

[Short circuit](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-device): $R$ near zero, so $I = U/R$ explodes — the stampede is division by almost-nothing. Open [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch): $R$ beyond [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring), so $I = U/R$ collapses to zero — the total blockade.

**Exercise 58.9 ★★.**

A delicate indicator lamp tolerates $20\,\mathrm{mA}$. What series [resistor](#def-g8-ohms-law-resistor) protects it on a $9\,\mathrm{V}$ [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), if the lamp itself takes about $2\,\mathrm{V}$ of the share? ([Resistor](#def-g8-ohms-law-resistor)’s share first, then $R$.)

**Solution of Exercise 58.9.**

The [resistor](#def-g8-ohms-law-resistor) must take $9 - 2 = 7\,\mathrm{V}$ at $0.02\,\mathrm{A}$: $R = 7 \div 0.02 = 350\,\Omega$.

**Exercise 58.10 ★★.**

A filament lamp’s portrait, measured: ($1\,\mathrm{V}$, $0.25\,\mathrm{A}$), ($4\,\mathrm{V}$, $0.5\,\mathrm{A}$). Show that no single $R$ fits both points, and tell the physical story behind the bending.

**Solution of Exercise 58.10.**

First point: $R = 1 \div 0.25 = 4\,\Omega$; second: $4 \div
0.5 = 8\,\Omega$ — no single value fits. The story: the growing current [heats](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) the filament, and hot metal resists more; the portrait bends because the component changes as it works.

**Exercise 58.11 ★★.**

Two [resistors](#def-g8-ohms-law-resistor) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series), $10\,\Omega$ and $20\,\Omega$, on $6\,\mathrm{V}$. Using the series laws plus Ohm’s law on each: find the loop’s current and each [resistor](#def-g8-ohms-law-resistor)’s [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage). (Hint: one current $I$ satisfies $6 = 10 I + 20 I$.)

**Solution of Exercise 58.11.**

One loop, one $I$: the two drops sum to the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) push, $6 = 10 I + 20 I = 30 I$, so $I = 0.2\,\mathrm{A}$. Then $U_{10}
= 10 \times 0.2 = 2\,\mathrm{V}$ and $U_{20} = 20 \times 0.2 =
4\,\mathrm{V}$ — summing to $6\,\mathrm{V}$, as the loop law demands.

**Exercise 58.12 ★★★.**

Same two [resistors](#def-g8-ohms-law-resistor), now [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) on $6\,\mathrm{V}$. Find each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s current, the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) total, and then — the elegant finish — the single [resistance](#def-g8-ohms-law-resistance) $R$ that would draw that same total from $6\,\mathrm{V}$. Compare it with both [resistors](#def-g8-ohms-law-resistor) and explain, in road-language, why the team of two resists *less* than either member alone.

**Solution of Exercise 58.12.**

Each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) spans $6\,\mathrm{V}$: $I_{10} = 6 \div 10 =
0.6\,\mathrm{A}$, $I_{20} = 6 \div 20 = 0.3\,\mathrm{A}$; total $0.9\,\mathrm{A}$. The stand-in: $R = 6 \div 0.9 \approx
6.7\,\Omega$ — *less* than either member. In road-language: two roads between the same towns carry more traffic than either alone; every added parallel road eases the passage, so the team’s opposition falls below its weakest member’s.

## 58.5 Problem: The Quality Control Bench

**Problem 58.1.**

Weekend problem — inspection day at the resistor factory; portraits, tolerances and one component that is no resistor at all

The factory’s quality bench tests the day’s production — and one impostor. You run the meters.

**Part I — Calibrating the bench.**

1. Sketch (or describe) the test circuit: supply, [ammeter](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-ammeter) , the component under test, [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) — who is [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) , who across?
2. Sample one, at $4.5\,\mathrm{V}$ , passes $0.15\,\mathrm{A}$ . Its [resistance](#def-g8-ohms-law-resistance) ?
3. The label claims $33\,\Omega$ , tolerance five percent. Compute the acceptable band, and rule on sample one. (Five percent of $33$ first.)
4. Sample two, labeled $150\,\Omega$ : predict its current at $4.5\,\mathrm{V}$ , so the bench knows what to expect (one decimal, in $\mathrm{mA}$ ).

**Part II — Full portraits.**

5. Sample three’s table: $(1.5, 0.05)$ , $(3.0, 0.10)$ , $(4.5, 0.15)$ , $(6.0, 0.20)$ — [volts](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) , [amperes](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity) . Fit $R$ , citing the feature of the numbers that permits a single value.
6. Sample four’s table: $(1.5, 0.30)$ , $(3.0, 0.45)$ , $(4.5, 0.54)$ , $(6.0, 0.60)$ . Show the impostor: no single $R$ , and the drift’s direction.
7. Sample four is, in fact, a small filament lamp from the next production line. Tell its bending story — and why the bench’s straight-line test is precisely a resistor-detector.
8. The bench’s rule book says: “every portrait must be taken at steady [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) .” Which clause of Ohm’s law is the rule protecting?

**Part III — Applications department.**

9. An order asks for a [resistor](#def-g8-ohms-law-resistor) limiting current to $30\,\mathrm{mA}$ on a $12\,\mathrm{V}$ supply (the load’s own share negligible). Compute the value to quote.
10. A customer complains their $25\,\Omega$ heater coil “only” draws $9.2\,\mathrm{A}$ while the [fuse](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-fuse) allows $10$ . Compute what supply [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) their complaint implies, and reassure them with the arithmetic.
11. The prototype lab requests the factory’s thickest, shortest copper jumper “with as close to zero [ohms](#def-g8-ohms-law-resistance) as possible”. What circuit role is that jumper born for — and near what everyday villain does its near-zero [resistance](#def-g8-ohms-law-resistance) place it if misused?
12. Close the inspection log: three sentences — the law, its portrait, and the one condition under which a component may sign it.

**Solution of Problem 58.1.**

**1.** Supply, [ammeter](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-ammeter) and component in one series loop — the tollbooth in the road; the [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) across the component — the surveyor aside. **2.** $R = 4.5 \div 0.15 = 30\,\Omega$. **3.** Five percent of $33$ is about $1.65$: acceptable from $31.35$ to $34.65\,\Omega$. Sample one’s $30\,\Omega$ falls outside: rejected. **4.** $I = 4.5 \div 150 = 0.03\,\mathrm{A} =
30.0\,\mathrm{mA}$. **5.** Every row returns $R = U/I = 30\,\Omega$ ($1.5
\div 0.05$, $3.0 \div 0.10$, …): constant ratio, straight-line portrait — one value fits all: $30\,\Omega$. **6.** The ratios run $5$, $6.7$, $8.3$, $10\,\Omega$: climbing with every step — [resistance](#def-g8-ohms-law-resistance) growing as the current grows; no single $R$ exists. **7.** As the current grows the filament [heats](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) toward glowing, and hot metal resists more — the portrait bends upward. The bench’s straight-line test thus admits exactly the components whose $R$ stays put: it is, by construction, a resistor-detector. **8.** The [steady-temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) clause: Ohm’s law promises proportionality *at steady [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature)* — warm a component mid-portrait and even an honest [resistor](#def-g8-ohms-law-resistor) drifts. **9.** $R = 12 \div 0.03 = 400\,\Omega$. **10.** $U = R I = 25 \times 9.2 = 230\,\mathrm{V}$ — exactly the mains: the coil draws precisely what Ohm’s law allots it, and the [fuse](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-fuse)’s margin is working as designed. Nothing is “only”; all is arithmetic. **11.** The jumper is born to be a connecting wire — the level road, spending no push. Misused across a supply, its near-zero [ohms](#def-g8-ohms-law-resistance) make it the perfect [short circuit](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-device): the villain was always just a very good wire in the wrong place. **12.** For example: “The law: $U = R I$, push equals opposition times march. Its portrait: a straight line through the origin, steeper for stronger opposers. And only a component at steady [temperature](https://one-course.com/books/physics/1/en/chapter/15-temperature-and-thermometers#def-g3-temperature-thermometers-temperature) may sign it — [heat](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) rewrites [resistance](#def-g8-ohms-law-resistance), and portraits taken feverish prove nothing.”