---
title: "Circuits: Loops, Series, Parallel"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 47
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/47-circuits-loops-series-parallel
---

# Chapter 47 — Circuits: Loops, Series, Parallel

Four years of circuits have filled your toolbox: loops, diagrams, the two great wirings. But one character has worked namelessly all along — the something that flows. This year it takes its name, its direction, and its strangest property: it is never used up. With the current on stage, series and parallel stop being recipes and become laws.

## 47.1 The current takes the stage

**Definition 47.1 (Electric current).**

The *electric current* is the flow that runs around a [closed circuit](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-circuit): a crowd of charged particles — unimaginably many, unimaginably small, cousins of the molecular world — set marching through the metal of the wires by the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s push. No loop, no march: the current exists only while the road is closed. (What exactly the marchers are, and how a metal lets them through, is a fine story reserved for the last years of this book.)

**Proposition 47.2 (The conventional direction).**

By worldwide convention, [circuit diagrams](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#def-g6-circuits-and-safety-diagram) mark the current as flowing *outside the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) from the $+$ [terminal](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), around the circuit, to the $-$ [terminal](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)*. Every arrow on every diagram in the world obeys this convention — chosen long before anyone could ask the marchers themselves. (When physics finally could ask, the answer held a famous small joke, saved for a later year.)

![The convention every electrician on Earth shares: current arrows leave the battery’s +, tour the loop, and return by -.](https://one-course.com/images/onecourse/chapters/physics-1/g7-circuits-series-parallel/fig-dd8903bcf406.svg)

*The convention every electrician on Earth shares: current arrows leave the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s $+$, tour the loop, and return by $-$.*

**Proposition 47.3 (The current is not consumed).**

Devices do not eat the current. Whatever current enters a lamp, [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor) or [buzzer](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor) leaves it entire, marching on: what the device takes is *[energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* from the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s store, delivered by the passing march. In any series loop, one and the same current therefore flows through every device — as much arrives back at the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) as ever left it.

**Example 47.4 (The misconception on trial).**

The tempting error: “the first [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) uses up some current, so the second must glow weaker.” The circuit itself refutes it. Wire two *identical* [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) and look: both glow with exactly equal brightness — and swapping them changes nothing. If current were consumed on the way, [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) two should always be the dim one, whichever [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) sits there. It never is. The current arrives whole; what the pair shares is the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s *push*, which is why both are dimmer than a lone [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) — equally dimmer.

## 47.2 The two families, as laws

**Proposition 47.5 (The series law).**

In a [series circuit](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) — one single loop — the same current flows through every device: one road, one march, no exceptions. Consequences, now explained: one gap stops the march everywhere at once; identical devices share the work equally, wherever they sit; and the order of devices on the loop cannot matter to a march that passes through all of them alike.

**Proposition 47.6 (The parallel law).**

At a [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), the current *divides*: part of the march takes each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), and the parts rejoin, complete, at the far [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — nothing lost, nothing gained. Each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) receives the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s full push, so each device works as if alone; and a blocked [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) merely sends nobody, while the other [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) march on.

![The parallel law drawn: the march divides at the first junction, each part tours its branch, and the full march reassembles at the second.](https://one-course.com/images/onecourse/chapters/physics-1/g7-circuits-series-parallel/fig-156c486b9729.svg)

*The parallel law drawn: the march divides at the first [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), each part tours its [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), and the full march reassembles at the second.*

**Example 47.7 (Reading brightness like an electrician).**

One [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), identical [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb). Alone: full brightness — the reference. Two [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series): both equally dim — one march, shared push. Two [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel): both fully bright — each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) enjoys the whole push (and the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) empties roughly twice as fast: nothing is free). Three [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series): dimmer still, all equal. The electrician’s glance: count the loop-mates for dimness, count the [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) for [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) bills.

**Method 47.8 (Predicting any lamp’s fate).**

Given a diagram, for each lamp:

1. *Existence* : is there at least one unbroken loop from $+$ through this lamp to $-$ ? No loop, no [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) .
2. *Company on the road* : which devices share *every* loop through this lamp? Those are its series companions — more companions, dimmer glow.
3. *[Branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) neighbors* : which devices sit on other [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) ? They neither dim this lamp nor darken with it.
4. *[Switches](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch)* : an open [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) kills exactly the loops it sits on — re-run step 1 with it removed.

**Remark 47.9 (Why the misconception dies hard).**

“Something must get used up — the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) empties!” True: the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s stored *[energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy)* is spent, handed to lamps as [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) and warmth by the passing march. But the marchers themselves circulate unharmed, like ski-lift chairs delivering skiers uphill all day without a chair ever being consumed. Keep the two bookkeepings apart — current circulates, [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) is delivered — and every circuit in this book stays honest. The instrument that counts the march itself joins us next year.

## 47.3 Exercises

**Exercise 47.1 ★.**

What is the [electric current](#def-g7-circuits-series-parallel-current)? When does it exist in a circuit, and when not?

**Solution of Exercise 47.1.**

The flow of charged particles marching through the circuit under the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s push. It exists only while at least one closed loop joins the two [terminals](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery); open the loop and the march stops everywhere at once.

**Exercise 47.2 ★.**

State the conventional direction of the current. On a diagram of a [one-bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) loop, where do the arrows point?

**Solution of Exercise 47.2.**

Outside the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), from $+$ around the circuit to $-$. On the [one-bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) loop the arrows leave $+$, pass through the [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb), and return to $-$.

**Exercise 47.3 ★.**

Two identical [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) glow exactly equally. What popular error does this equality refute, and how?

**Solution of Exercise 47.3.**

The error that devices consume current, so “downstream” [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) should glow weaker. Equal brightness — surviving every swap — shows the same whole current passes through both: nothing is eaten en route.

**Exercise 47.4 ★.**

In a series loop with a [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor) and a lamp, the current through the [motor](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor) is compared with the current through the lamp. What does the series law say?

**Solution of Exercise 47.4.**

They are one and the same current: a single loop carries a single march through every device on it.

**Exercise 47.5 ★.**

What happens to the current at a [parallel circuit](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s first [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)? And at the second? What is true of each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s share of the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s push?

**Solution of Exercise 47.5.**

It divides — part of the march takes each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — and at the second [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) the parts rejoin, complete. Every [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) enjoys the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s full push.

**Exercise 47.6 ★.**

One [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), identical [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb): rank the brightness of a [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) — alone; with one series companion; with one parallel neighbor; with two series companions.

**Solution of Exercise 47.6.**

Alone $=$ with one parallel neighbor (full brightness), then one series companion (dim), then two series companions (dimmer still).

**Exercise 47.7 ★.**

Which arrangement empties the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) faster: two [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series), or the same two [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)? Why, in march-language?

**Solution of Exercise 47.7.**

Parallel: two [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) each drawing a full march doubles the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s total delivery, while the series pair sends one throttled march through both. Full brightness is paid for at the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery).

**Exercise 47.8 ★.**

Explain the ski-lift picture of [Remark 47.9](#rem-g7-circuits-series-parallel-whyhard): what plays the chairs, what plays the skiers, and what is truly used up?

**Solution of Exercise 47.8.**

The chairs are the charged marchers, circulating endlessly and never consumed; the skiers are the [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy), loaded at the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) and dropped off at the lamps as [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) and warmth. What runs out is the resort’s stock of skiers — the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery)’s stored [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy).

**Exercise 47.9 ★★.**

A diagram: [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch), then lamp A; after A a [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) sends two [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — lamp B on one, lamp C on the other — rejoining before returning to the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery). Using [Method 47.8](#met-g7-circuits-series-parallel-predict): who are A’s series companions? What does A’s glow do when C’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) is cut?

**Solution of Exercise 47.9.**

Every loop through A passes the [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) and then either B or C — so A’s series companions are the [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) and the *pair* of [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) as a whole, but neither B nor C alone. Cutting C’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) sends the whole march through B: A keeps glowing (a little differently — one road instead of two — as next year’s instruments will [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring)).

**Exercise 47.10 ★★.**

Decorative [lights](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) ask for a design: ten [bulbs](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) that all glow at full, independent brightness from one supply — but the whole set must obey a single master [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch). Draw or describe the wiring, citing both laws.

**Solution of Exercise 47.10.**

One master switch on the common road just after the supply, then ten parallel [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), one [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) each. The parallel law grants full, independent brightness to every [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb); the series placement of the master — on the road every loop shares — lets one [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) command all ten.

**Exercise 47.11 ★★.**

In the [two-bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) [parallel circuit](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) 1’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) is thick copper wire while [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) 2’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) includes a long, thin wire. [Bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) 2 glows a little weaker. What does this suggest about how a [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s road itself can throttle its share of the march? (Next year gives this throttling a name and a law.)

**Solution of Exercise 47.11.**

The [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s division is not an equal split: a [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s own road can resist the march, and the long thin wire throttles its [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s share — fewer marchers per second, weaker glow. Roads themselves fight the current: the idea behind next year’s resistance.

**Exercise 47.12 ★★★.**

A [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) feeds two parallel [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel): [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) one holds lamp X alone; [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) two holds lamps Y and Z [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series). Compare the three lamps’ brightnesses, in order, with reasons from both laws. Then predict every lamp’s fate if Z burns out — and if instead X burns out.

**Solution of Exercise 47.12.**

X, alone on its [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), gets the full push: brightest. Y and Z share their [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)’s push [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series): equally dim, both below X. If Z burns out, its loop is broken: Y darkens with it — X shines on, indifferent. If X burns out, only X’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) dies: Y and Z continue, dim as before.

## 47.4 Problem: The Lighthouse Keeper’s Panel

**Problem 47.1.**

Weekend problem — rewiring the old lighthouse; one battery bank, many duties; the keeper’s night of faults

The lighthouse’s [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) bank must feed: the great lamp, a foghorn (a mighty [buzzer](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#ex-g6-circuits-and-safety-motor)), the keeper’s reading [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source), and a staircase [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) — and survive the keeper’s troubleshooting night.

**Part I — Design.**

1. The great lamp must never dim because something else switches on. What wiring does this demand for the whole installation, and why would any series pairing betray the ships?
2. Each duty needs its own [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) , and the keeper wants one master cut-off for storms. Place all five [switches](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) .
3. The foghorn and the staircase [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) must never be prevented from working by each other’s faults. Is this already guaranteed? By which law?
4. Sketch or describe the full diagram: bank, master [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) , four [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) , four [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) [switches](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) .

**Part II — The keeper’s questions.**

5. The keeper worries: “Four duties drinking at once — does each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) still get the bank’s full push?” Answer with the parallel law.
6. “And is current being used up as it works through the great lamp?” Set the keeper straight, with the ski-lift if it helps.
7. On a foggy night everything runs at once. What is the cost of parallel generosity, and where in the lighthouse is it paid?
8. The reading [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) alone is lit; measure-minded, the keeper follows the march: describe its full round trip, naming every component it passes and every one it avoids.

**Part III — The night of faults.**

9. Midnight: the staircase [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) burns out. What else goes dark? Which law answers?
10. One o’clock: a clumsy repair leaves a stretch of bare wire joining the two [terminals](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) of the bank directly, upstream of every [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) . Name the event, and explain why every duty in the lighthouse now fails at once — where does the march go?
11. Two o’clock, fault cleared: the keeper wants the great lamp protected so that such a shortcut can never again silence it. A second, independent [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) bank is available. Propose the wiring.
12. Dawn: write the keeper’s log — three sentences: which law kept ships safe tonight, which misconception the night refuted, and which single wire caused the worst hour.

**Solution of Problem 47.1.**

**1.** Four parallel [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), one duty each: [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) works at full strength whatever the others do. Any series pairing would let a reading lamp’s [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) or a burnt staircase [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb) dim or darken the great lamp — unforgivable. **2.** The master on the common road just after the bank; each of the four [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) [switches](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) with its own duty, inside its [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel). **3.** Yes — the parallel law: a gap in one [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) stops only 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 others march on. **4.** Bank, master [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch), then a [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) into four [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — (great lamp $+$ [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch)), (foghorn $+$ [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch)), (reading [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) $+$ [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch)), (staircase [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) $+$ [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch)) — rejoining and returning to the bank. **5.** Yes: every [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) spans the bank’s two sides, so each receives the full push regardless of how many neighbors drink alongside. **6.** No: the current through the lamp leaves it whole — the chairs circulate uneaten. What the lamp takes is [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) from the bank’s store, delivered by the passing march. **7.** Four full marches at once: the bank’s [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy) drains four duties fast. The bill is paid in the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) room — in stored [energy](https://one-course.com/books/physics/1/en/chapter/29-energy-in-everyday-life#def-g5-everyday-energy-energy), never in lost current. **8.** Out of the bank’s $+$, through the master [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch), to the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel); into the reading [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source)’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), through its closed [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) and the lamp; past the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) where the other three (dark) [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) rejoin; back to the bank’s $-$. It avoids the great lamp, foghorn and staircase [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) entirely. **9.** Nothing else — only the staircase [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) dies with its [bulb](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-bulb): the parallel law’s independence. **10.** A [short circuit](https://one-course.com/books/physics/1/en/chapter/41-electric-circuits-and-safety#def-g6-circuits-and-safety-short) upstream of every [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel). The march abandons every working road for the effortless bare shortcut: the lamps and horn starve while the shortcut wire carries a wild, heating rush — every duty fails at once until the wire is removed. **11.** Give the great lamp its own independent loop on the second bank: bank two, its own [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch), the great lamp — sharing no wire with the general panel. A shortcut across bank one then silences everything *except* the ships’ [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source). **12.** For example: “The parallel law kept each duty sovereign tonight. The night refuted, once more, the belief that lamps devour current — the march returned whole even as the banks drained. And one bare wire, shorting the bank, cost us our worst hour: every road empty while the shortcut burned.”