---
title: "Alternating Current"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 67
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/67-alternating-current
---

# Chapter 67 — Alternating Current

Every circuit you have ever built ran on a [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery): current marching steadily from $+$ to $-$, dutiful as a clock’s hand. But the sockets in your walls deliver something wilder — a current that reverses direction a hundred times each second, its [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) swinging ceaselessly between plus and minus. Meet the [alternating current](#def-g9-alternating-current-dcac): the form in which virtually all the world’s electricity is born, shipped and delivered.

## 67.1 Two kinds of current

**Definition 67.1 (Direct and alternating current).**

A *direct current* (DC) flows always in the same direction, driven by a steady [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) gift, constant until the store runs down. An *alternating current* (AC) reverses direction periodically, driven by a [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) that swings smoothly from positive to negative and back, over and over — the socket’s regime, and the alternator’s (next chapter tells whose).

![Two voltages against time: the battery’s flat, faithful line — and the mains’ endless swing between plus and minus, twenty milliseconds per full cycle.](https://one-course.com/images/onecourse/chapters/physics-1/g9-alternating-current/fig-aad290a3a59d.svg)

*Two [voltages](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) against time: the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) flat, faithful line — and the mains’ endless swing between plus and minus, twenty milliseconds per full cycle.*

**Definition 67.2 (Period and frequency).**

The *period* $T$ of an alternating [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) is the [duration](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#def-g2-measuring-time-duration) of one complete cycle — one full swing up, down and back, in seconds. Its *[frequency](https://one-course.com/books/physics/1/en/chapter/62-sound-speed-pitch-loudness#def-g8-sound-pitch-loudness-frequency)* $f$ is the number of cycles per second, in [hertz](https://one-course.com/books/physics/1/en/chapter/62-sound-speed-pitch-loudness#def-g8-sound-pitch-loudness-frequency) — [sound](https://one-course.com/books/physics/1/en/chapter/19-sound-around-us#def-g3-sound-around-us-sound)’s [unit](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring), serving again — and the two are each other’s reciprocals:

$$
f = \frac{1}{T}, \qquad T = \frac{1}{f}.
$$

**Proposition 67.3 (The mains’ vital statistics).**

Across most of the world the mains alternates at $f = 50\,\mathrm{Hz}$ — fifty full cycles per second, a [period](#def-g9-alternating-current-period) of $T = 1/50 = 0.02\,\mathrm{s}$: twenty milliseconds per swing (some regions run at sixty). The current in your reading lamp’s cord reverses direction a hundred times each second — twice per cycle — and has never once marched steadily from one [terminal](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) to the other.

**Example 67.4 (Frequency arithmetic).**

The reciprocal pair computes both ways. A generator cycling at $60\,\mathrm{Hz}$: $T = 1/60 \approx 0.017\,\mathrm{s}$ — seventeen milliseconds. An [oscilloscope](#met-g9-alternating-current-scope) shows a cycle lasting $0.04\,\mathrm{s}$: $f = 1/0.04 = 25\,\mathrm{Hz}$. A bicycle dynamo at slow pedaling makes $10\,\mathrm{Hz}$: [period](#def-g9-alternating-current-period) a tenth of a second — and its lamp visibly flickers, ten pulses to the second, until [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) smooths them beyond the eye’s noticing.

## 67.2 Reading the swing

**Method 67.5 (Reading an oscilloscope trace).**

The *oscilloscope* — or any analyzer app — draws [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) against time on a squared screen; each square’s worth in [volts](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) and in milliseconds is set by two dials. To harvest a trace:

1. count squares for one full cycle and multiply by the time-per-square: that is $T$ ; then $f = 1/T$ ;
2. count squares from the middle line to a crest and multiply by the volts-per-square: the *peak [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage)* $U_{\text{max}}$ ;
3. check the shape: mains and alternators draw the smooth, endlessly repeated wave of the figure — the *sinusoid* , mathematics’ name for the perfect swing.

**Proposition 67.6 (Effective voltage).**

What is “the” [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) of a swing that visits every value from $+U_{\text{max}}$ to $-U_{\text{max}}$ a hundred times a second? The honest single number is the *effective voltage* $U$: the steady DC [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) that would [heat](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) a [resistor](https://one-course.com/books/physics/1/en/chapter/58-resistance-and-ohms-law#def-g8-ohms-law-resistor) equally well. For the sinusoid it sits well below the crest — $U_{\text{max}}$ is about $1.4$ times $U$ — and it is what every AC [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) is built to read. The famous “$230\,\mathrm{V}$” of the socket is the effective value; the crests actually reach about $325\,\mathrm{V}$, fifty times a second, each way.

**Example 67.7 (Which devices care).**

The [heating effect](https://one-course.com/books/physics/1/en/chapter/54-electric-current-and-its-effects#prop-g8-electric-current-effects-three) works fine either way — a filament or kettle coil warms regardless of the march’s direction, which is why lamps and heaters drink AC without complaint. But electronics — phones, laptops, anything with chips — need steady DC: their charger bricks and hidden power supplies convert the socket’s swing into a battery-like constancy (feel the brick’s warmth: conversion pays its tax). And the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) world cannot charge on a swing either: charging is chemistry run backward, and chemistry insists on a consistent direction.

**Remark 67.8 (Why the world chose the swing).**

If batteries are so agreeably steady, why is civilization wired for AC? Two reasons, both waiting two chapters ahead: [alternating current](#def-g9-alternating-current-dcac) is what spinning machines *naturally make* — and, decisively, only AC feeds the transformer, the humble humming box that lets electricity travel a country at colossal [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) and enter your home at a tame one. The grid is alternating because the journey demands it; the next two chapters follow the current from spinning coil to kitchen wall.

**Remark 67.9 (The swing and the body).**

Safety’s old arithmetic gains one grim detail: the mains’ fifty swings a second lie near the rhythm that most easily disrupts the heart’s own electrical timing — alternating shocks are, [ampere](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity) for [ampere](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity), more dangerous than steady ones. Every childhood rule stands, doubled: the socket’s $230\,\mathrm{V}$ effective — $325\,\mathrm{V}$ at crest — is not merely a bigger [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery). It is a different, less forgiving animal.

## 67.3 Exercises

**Exercise 67.1 ★.**

Distinguish DC from AC in one sentence each, with each one’s household source.

**Solution of Exercise 67.1.**

DC flows always one way under a steady [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) regime. AC reverses periodically under a swinging [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — the socket’s regime.

**Exercise 67.2 ★.**

Define [period](#def-g9-alternating-current-period) and [frequency](https://one-course.com/books/physics/1/en/chapter/62-sound-speed-pitch-loudness#def-g8-sound-pitch-loudness-frequency) and write their reciprocal relation. Compute $T$ for $50\,\mathrm{Hz}$ and $f$ for $T = 0.01\,\mathrm{s}$.

**Solution of Exercise 67.2.**

$T$: the [duration](https://one-course.com/books/physics/1/en/chapter/8-measuring-time#def-g2-measuring-time-duration) of one full cycle; $f$: cycles per second; $f = 1/T$ and $T = 1/f$. For $50\,\mathrm{Hz}$: $T =
0.02\,\mathrm{s}$. For $T = 0.01\,\mathrm{s}$: $f = 100\,\mathrm{Hz}$.

**Exercise 67.3 ★.**

How many times per second does the mains current reverse direction at $50\,\mathrm{Hz}$? (Careful: reversals per cycle.)

**Solution of Exercise 67.3.**

A hundred times: two reversals in every one of the fifty cycles.

**Exercise 67.4 ★.**

A scope shows one cycle spanning $4$ squares at $5\,\mathrm{ms}$ per square. Find $T$ and $f$.

**Solution of Exercise 67.4.**

$T = 4 \times 5 = 20\,\mathrm{ms} = 0.02\,\mathrm{s}$; $f = 1/0.02
= 50\,\mathrm{Hz}$.

**Exercise 67.5 ★.**

The socket says $230\,\mathrm{V}$; the crest reaches $325\,\mathrm{V}$. Name the two values and the meaning of the smaller one.

**Solution of Exercise 67.5.**

$230\,\mathrm{V}$ is the [effective voltage](#prop-g9-alternating-current-effective) — the steady DC value that would [heat](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat) equally well, and what AC [voltmeters](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) read; $325\,\mathrm{V}$ is the peak, the crest the swing actually touches, each way, fifty times a second.

**Exercise 67.6 ★.**

Why do kettles accept AC happily while phone chargers must convert it? One effect, one chemistry.

**Solution of Exercise 67.6.**

Heating cares nothing for direction — the coil warms on any march, so kettles drink the swing directly. Charging is chemistry run backward, and chemistry demands one consistent direction: the brick converts the swing to steadiness first.

**Exercise 67.7 ★.**

Sketch (or describe) the two traces: 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), and an alternating [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) of [period](#def-g9-alternating-current-period) $20\,\mathrm{ms}$ and peak $9\,\mathrm{V}$. What single instrument-drawn feature tells them apart at a glance?

**Solution of Exercise 67.7.**

The [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery): a flat line at $9\,\mathrm{V}$. The alternating supply: a smooth wave swinging between $+9$ and $-9\,\mathrm{V}$, one cycle per $20\,\mathrm{ms}$. The glance-test: flat against wavy.

**Exercise 67.8 ★.**

Give the two reasons civilization wired itself for AC, as previewed by the chapter.

**Solution of Exercise 67.8.**

Spinning machines make AC naturally; and only AC feeds the transformer, without which electricity could not travel the country at high [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) and enter homes at a tame one.

**Exercise 67.9 ★★.**

A slow bicycle dynamo [lights](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) its lamp with visible flicker; pedaling harder steadies the glow. Explain both regimes with [frequency](https://one-course.com/books/physics/1/en/chapter/62-sound-speed-pitch-loudness#def-g8-sound-pitch-loudness-frequency) — and estimate the [frequency](https://one-course.com/books/physics/1/en/chapter/62-sound-speed-pitch-loudness#def-g8-sound-pitch-loudness-frequency) below which flicker annoys, given that cinema fools the eye at about $24$ pictures per second.

**Solution of Exercise 67.9.**

At low pedaling the dynamo’s few cycles per second pulse the lamp visibly — each crest a flash, each zero a dip; faster pedaling multiplies the pulses until the eye [fuses](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-fuse) them. Cinema’s $24$ pictures set the eye’s rough fusion rate: below about $25\,\mathrm{Hz}$ of light-pulses, flicker annoys; above, the glow reads steady.

**Exercise 67.10 ★★.**

An AC [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) across a socket reads $230\,\mathrm{V}$. A (theoretical) fearless [oscilloscope](#met-g9-alternating-current-scope) across the same socket shows crests at what value — and how many crests per second, counting both signs?

**Solution of Exercise 67.10.**

Crests at about $325\,\mathrm{V}$ — and a hundred of them per second, fifty positive and fifty negative.

**Exercise 67.11 ★★.**

A region’s grid runs at $60\,\mathrm{Hz}$ with effective $120\,\mathrm{V}$. Compute its [period](#def-g9-alternating-current-period) and approximate peak [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — and name which of your travel devices would object (consult [Example 67.7](#ex-g9-alternating-current-whocares); modern chargers read the fine print on their labels).

**Solution of Exercise 67.11.**

$T = 1/60 \approx 0.017\,\mathrm{s}$; peak $\approx 120 \times
1.4 \approx 170\,\mathrm{V}$. Heaters and lamps adjust indifferently; modern chargers read “100–240 V, 50/60 Hz” on their labels and convert happily — the objectors are old [single-voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) devices, and any motorized clock that counts the grid’s cycles: it would run fast on sixty.

**Exercise 67.12 ★★★.**

Old films of screens and fluorescent lamps sometimes show dark rolling bars — the camera’s frames catching the lamp’s brightness mid-swing. A mains lamp at $50\,\mathrm{Hz}$ brightens at every crest of *either* sign: how many brightness pulses per second? A camera at $25$ frames per second samples that pulsing: reason out why the beat between the two rhythms paints slow-moving bars, and why filming in a $60\,\mathrm{Hz}$ region changes the pattern.

**Solution of Exercise 67.12.**

Brightening at both crests gives $2 \times 50 = 100$ pulses per second. A $25$-frame camera samples every $0.04\,\mathrm{s}$ — four pulses per frame, but not aligned identically at every point of the rolling exposure: the slight mismatch between the camera’s rhythm and the lamp’s paints bands that creep across successive frames — a beat between two clocks. At $60\,\mathrm{Hz}$ ($120$ pulses), the mismatch against $25$ frames changes, and the bars change width and [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) — the pattern betrays the local grid.

## 67.4 Problem: The Oscilloscope Bench

**Problem 67.1.**

Weekend problem — qualification night on the oscilloscope bench; four mystery supplies, one screen; the technician’s ledger

Four unlabeled supplies must be identified from their traces. The scope’s dials: $5\,\mathrm{ms}$ per horizontal square, $2\,\mathrm{V}$ per vertical square (except where noted). You keep the ledger.

**Part I — The four traces.**

1. Supply A: a flat line, $2.25$ squares above the middle. Identify its kind and [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — and its likely household identity.
2. Supply B: a smooth wave, one cycle per $4$ squares, crests $3$ squares high. Kind, [period](#def-g9-alternating-current-period) , [frequency](https://one-course.com/books/physics/1/en/chapter/62-sound-speed-pitch-loudness#def-g8-sound-pitch-loudness-frequency) , peak [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) ?
3. Supply C: a flat line *below* the middle by $2.25$ squares. What is it, and what tiny error explains it?
4. Supply D: a smooth wave, one cycle per $2$ squares, crests $1.5$ squares. Full statistics — and why D’s lamp would flicker less than B’s if both fed one.

**Part II — The technician’s questions.**

5. An AC [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) across supply B reads about $4.2\,\mathrm{V}$ , though its crests hit $6\,\mathrm{V}$ . Reconcile the two numbers.
6. Which of the four supplies could charge a phone directly, and what must stand between the others and its [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) ?
7. The bench’s warning plate: “AC bites worse, [volt](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) for [volt](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) .” Restate the reason.
8. A trainee asks why the scope draws B’s wave so smoothly rounded rather than zigzag. Name the shape and its origin (what kind of machine, next chapter, draws it naturally?).

**Part III — The grid’s own trace.** A safely isolated demonstrator shows the mains, at $100\,\mathrm{V}$ and $5\,\mathrm{ms}$ per square.

9. Predict the trace: squares per cycle, and crest height in squares.
10. From the trace, recover $f$ and the [effective voltage](#prop-g9-alternating-current-effective) — and check both against the socket’s famous label.
11. The demonstrator is switched to a [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) bank of equal effective heating power. Describe the new trace in one line.
12. Close the ledger with the technician’s three-line summary: what the flat line means, what the wave means, and which single number lets the two be fairly compared.

**Solution of Problem 67.1.**

**1.** DC at $2.25 \times 2 = 4.5\,\mathrm{V}$: the flat pocket [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery). **2.** AC; $T = 20\,\mathrm{ms}$, $f = 50\,\mathrm{Hz}$; peak $3 \times 2 = 6\,\mathrm{V}$. **3.** DC at $4.5\,\mathrm{V}$ connected *backward* — the leads swapped: harmless on a scope, instructive in the ledger. **4.** AC; $T = 10\,\mathrm{ms}$, $f = 100\,\mathrm{Hz}$; peak $3\,\mathrm{V}$. Its hundred-per-second doubled pulses sit beyond the eye’s fusion rate more comfortably than B’s fifty-cycle flicker. **5.** The [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) reports the effective value — the DC-equivalent for heating — which for a sinusoid sits at about the peak divided by $1.4$: $6 \div 1.4 \approx
4.3\,\mathrm{V}$. Both numbers describe one swing. **6.** Supply A alone (right way round — C after re-plugging). The AC supplies need a converter — the charger’s rectifying brick — between them and any [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery). **7.** The mains’ fifty swings a second lie near the heart’s own electrical rhythm and can disrupt it: [ampere](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity) for [ampere](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity), the swing endangers more than the steady march. **8.** The sinusoid — the perfect swing, drawn naturally by a coil turning at constant [speed](https://one-course.com/books/physics/1/en/chapter/52-motion-graphs-and-average-speed#def-g7-motion-average-speed-formula) in a magnetic field: the alternator of the next chapter. **9.** One cycle per $4$ squares ($20\,\mathrm{ms}$); crests $325 \div 100 = 3.25$ squares above and below. **10.** $f = 1/0.02 = 50\,\mathrm{Hz}$; effective $\approx
325 \div 1.4 \approx 230\,\mathrm{V}$ — the label, recovered from the screen. **11.** A flat line at $230\,\mathrm{V}$ — steadiness of equal heating power. **12.** For example: “A flat line is a [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) constancy — one direction, one value. A wave is the grid’s swing — direction and value in ceaseless cycle. And the fair exchange rate between them is the [effective voltage](#prop-g9-alternating-current-effective): equal [heat](https://one-course.com/books/physics/1/en/chapter/34-heat-and-insulation#def-g5-heat-and-insulation-heat), equal worth.”