---
title: "Current and Voltage Laws in Circuits"
book: "Primary & Middle School Physics"
subject: physics
language: en
chapter: 57
exercises: 12
source: https://one-course.com/books/physics/1/en/chapter/57-current-and-voltage-laws-in-circuits
---

# Chapter 57 — Current and Voltage Laws in Circuits

Two meters, two quantities — and now, at last, the constitution. Every circuit you will ever build, from a torch to a city grid, obeys four short laws: two about the current’s bookkeeping, two about the [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage)’s. Armed with them and a little algebra, you can compute readings you have not yet measured — and catch any measurement that lies.

## 57.1 The current laws

**Proposition 57.1 (Current law for a series loop).**

Along a single loop, the [intensity](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity) is the same everywhere:

$$
I_{\text{battery}} = I_{\text{lamp 1}} = I_{\text{lamp 2}} =
\dots
$$

One road, one march — last chapter’s tollbooths, now a law of the constitution.

**Proposition 57.2 (The junction law).**

At any [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel), the intensities flowing in together equal the intensities flowing out together: for a main road splitting into two [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel),

$$
I = I_1 + I_2 .
$$

Charge is neither created, destroyed nor stored at a fork: whatever arrives, leaves. (True for any number of [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — sum the arrivals, sum the departures, and the books balance.)

![The junction law with numbers: 0.50\, A arrives at the fork, 0.30 + 0.20 leave along the branches — and 0.50\, A reassembles at the far junction.](https://one-course.com/images/onecourse/chapters/physics-1/g8-circuit-laws/fig-adb2d0de4698.svg)

*The [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law with numbers: $0.50\,\mathrm{A}$ arrives at the fork, $0.30 + 0.20$ leave along the [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — and $0.50\,\mathrm{A}$ reassembles at the far [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel).*

## 57.2 The voltage laws

**Proposition 57.3 (Voltage law for a series loop).**

Around a series loop, the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) is shared among the devices, and the shares add up exactly:

$$
U_{\text{battery}} = U_1 + U_2 + \dots
$$

The waterfall [image](https://one-course.com/books/physics/1/en/chapter/32-mirrors-and-reflection#prop-g5-mirrors-reflection-image) keeps the books: the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) lifts the march through its full height, and the devices spend that height between them, drop by drop — with good wires spending none.

**Proposition 57.4 (Voltage law for parallel branches).**

Devices [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) all span the same two [junctions](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — so one [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) serves them all:

$$
U_1 = U_2 = U_{\text{across the pair of junctions}} .
$$

Each [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) enjoys the full push between the [junctions](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) — which is why parallel lamps shine at full nominal brightness, and why your house’s every socket offers the same $230\,\mathrm{V}$.

![The series voltage law with numbers: the battery’s 4.5\, V is spent in two unequal shares — 2.1 + 2.4 = 4.5, to the decimal.](https://one-course.com/images/onecourse/chapters/physics-1/g8-circuit-laws/fig-1a26190cc4ff.svg)

*The series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law with numbers: the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) $4.5\,\mathrm{V}$ is spent in two unequal shares — $2.1 + 2.4 =
4.5$, to the decimal.*

**Example 57.5 (Why unequal shares?).**

Two different lamps [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) carry the same current (law one) yet split the [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) unequally — $2.1\,\mathrm{V}$ against $2.4\,\mathrm{V}$. The bigger share goes to the lamp that opposes the march more: the harder worker takes the larger drop. What exactly “opposes more” means, and how to [measure](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) it, is the next chapter’s whole subject — the laws of this chapter are its stage.

## 57.3 Computing before measuring

**Method 57.6 (Solving a circuit with the four laws).**

Given a circuit with some readings known:

1. draw the diagram and mark every known $I$ and $U$ ;
2. write the applicable laws as equations — [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) sums for currents, loop sums for [voltages](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) , sameness for series $I$ and parallel $U$ ;
3. solve for the unknowns (your algebra course’s equations, earning their keep);
4. sanity-check: no lamp receiving more than the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) offers, no [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) current exceeding the main road’s.

**Example 57.7 (Worked: the mixed circuit).**

A $6\,\mathrm{V}$ [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) feeds lamp A [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) with a parallel pair (B and C). Measured: $I_{\text{battery}} = 0.5\,\mathrm{A}$, $I_B = 0.3\,\mathrm{A}$, $U_A = 2.5\,\mathrm{V}$. Compute the rest. [Junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law: $I_C = 0.5 - 0.3 = 0.2\,\mathrm{A}$. Series current law: lamp A carries the full $0.5\,\mathrm{A}$. Loop [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law: the pair receives $U_{\text{pair}} = 6 - 2.5 = 3.5\,\mathrm{V}$; parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law: $U_B = U_C = 3.5\,\mathrm{V}$. Every meter reading in the circuit, now known — four laws, one line of algebra each.

**Example 57.8 (The laws as lie detectors).**

A lab report claims: [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) $4.5\,\mathrm{V}$; two series lamps at $2.1\,\mathrm{V}$ and $2.9\,\mathrm{V}$. The loop law adds the shares: $5.0\,\mathrm{V}$ spent from a $4.5\,\mathrm{V}$ purse — impossible; someone misread a dial. Another report: [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) in $0.40\,\mathrm{A}$, [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) out $0.25\,\mathrm{A}$ and $0.20\,\mathrm{A}$ — the fork “creates” $0.05\,\mathrm{A}$: rejected. Before the laws, wrong readings looked as good as right ones; now the constitution audits every page.

**Remark 57.9 (Batteries under the same laws).**

The nose-to-tail rule is the series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law wearing its oldest clothes: batteries [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) add their [voltages](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) ($1.5 +
1.5 + 1.5 = 4.5\,\mathrm{V}$ — the flat [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) secret), and one reversed cell *subtracts*, its push spent fighting the others. The laws govern givers and spenders alike; only the sign of their contribution differs.

## 57.4 Exercises

**Exercise 57.1 ★.**

State the four laws — two for current, two for [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — each in one line.

**Solution of Exercise 57.1.**

Series current law: one loop, one [intensity](https://one-course.com/books/physics/1/en/chapter/55-current-intensity-and-the-ammeter#def-g8-intensity-ammeter-intensity) everywhere. [Junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law: currents in equal currents out. Series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law: the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) equals the sum of the devices’ shares around the loop. Parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law: [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) spanning the same [junctions](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) share one and the same [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage).

**Exercise 57.2 ★.**

A [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) receives $0.8\,\mathrm{A}$ and sends $0.55\,\mathrm{A}$ down one [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel). The other [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) carries?

**Solution of Exercise 57.2.**

$0.8 - 0.55 = 0.25\,\mathrm{A}$.

**Exercise 57.3 ★.**

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) feeds three series lamps; two of them take $2.5\,\mathrm{V}$ and $3.8\,\mathrm{V}$. The third’s share?

**Solution of Exercise 57.3.**

$9 - 2.5 - 3.8 = 2.7\,\mathrm{V}$.

**Exercise 57.4 ★.**

Two lamps [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) across a $12\,\mathrm{V}$ [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery): each lamp’s [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage)? What everyday fact about house sockets is the same law?

**Solution of Exercise 57.4.**

$12\,\mathrm{V}$ each — the parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law. The same law puts the identical $230\,\mathrm{V}$ on every socket of the house’s parallel wiring.

**Exercise 57.5 ★.**

In the worked mixed circuit, which single further measurement would have been redundant, and why? (Pick any, justify by a law.)

**Solution of Exercise 57.5.**

For instance $I_C$: once $I_{\text{battery}}$ and $I_B$ are known, the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law fixes $I_C = 0.2\,\mathrm{A}$ — any meter sent there could only confirm (likewise $U_C$, fixed by the parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law once $U_B$ is known).

**Exercise 57.6 ★.**

Two different series lamps carry the same current but split the [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) $1.8$ against $2.7\,\mathrm{V}$. Which lamp opposes the march more? What chapter-to-come [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) that opposition?

**Solution of Exercise 57.6.**

The $2.7\,\mathrm{V}$ lamp: at equal current, the larger [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) share marks the stronger opposer. The next chapter [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) that opposition — resistance.

**Exercise 57.7 ★.**

Audit: [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) $6\,\mathrm{V}$; series lamps measured $2.2\,\mathrm{V}$, $1.9\,\mathrm{V}$, $1.4\,\mathrm{V}$. Do the books balance within a meter’s honesty?

**Solution of Exercise 57.7.**

$2.2 + 1.9 + 1.4 = 5.5\,\mathrm{V}$ against a $6\,\mathrm{V}$ purse: half a [volt](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) astray — beyond honest meter scatter for such readings; one dial was misread (or a share went unmeasured).

**Exercise 57.8 ★.**

Audit: main road $0.9\,\mathrm{A}$; three [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) $0.4\,\mathrm{A}$, $0.3\,\mathrm{A}$, $0.3\,\mathrm{A}$. Verdict?

**Solution of Exercise 57.8.**

$0.4 + 0.3 + 0.3 = 1.0\,\mathrm{A}$ leaving against $0.9\,\mathrm{A}$ arriving: the fork “creates” $0.1\,\mathrm{A}$ — convicted by the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law.

**Exercise 57.9 ★★.**

A $4.5\,\mathrm{V}$ [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), lamp D [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) with a [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) — then, [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) with lamp D, a [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) shows $4.5\,\mathrm{V}$ across the *open* [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) and $0\,\mathrm{V}$ across D. Show both readings obey the loop [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law.

**Solution of Exercise 57.9.**

The loop’s spending must 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) $4.5\,\mathrm{V}$: open [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) $4.5\,\mathrm{V}$ $+$ idle lamp $0\,\mathrm{V}$ $= 4.5$ — balanced. (After the click the shares trade: [switch](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#ex-g3-first-electric-circuit-switch) near zero, lamp near $4.5\,\mathrm{V}$ — balanced again.)

**Exercise 57.10 ★★.**

Three identical cells nose-to-tail [light](https://one-course.com/books/physics/1/en/chapter/3-light-and-shadows#def-g1-light-and-shadows-source) a lamp; a fourth, reversed, joins the queue. Total [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) before and after, by the series law with signs? What does the lamp do?

**Solution of Exercise 57.10.**

Before: $3 \times 1.5 = 4.5\,\mathrm{V}$. After: $4.5 - 1.5 =
3.0\,\mathrm{V}$ — the reversed cell subtracts. The lamp dims from its $4.5\,\mathrm{V}$ glow to a $3\,\mathrm{V}$ one.

**Exercise 57.11 ★★.**

A [two-branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) circuit: [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) one carries lamps E and F [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series); [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) two carries lamp G alone. [Battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) $6\,\mathrm{V}$; $U_E = 2.2\,\mathrm{V}$. Compute $U_F$ and $U_G$, citing a law for each step.

**Solution of Exercise 57.11.**

[Branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) one spans the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery): $U_E + U_F = 6\,\mathrm{V}$ (series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law along the [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel)), so $U_F = 6 - 2.2 =
3.8\,\mathrm{V}$. [Branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) two spans the same [junctions](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel): $U_G =
6\,\mathrm{V}$ (parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law).

**Exercise 57.12 ★★★.**

The complete audit: a $6\,\mathrm{V}$ [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery); lamp P [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) with a [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel); [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) Q and R rejoin and return. Measured: $I_P = 0.6\,\mathrm{A}$, $I_Q = 0.45\,\mathrm{A}$, $U_Q = 2.3\,\mathrm{V}$. Compute every remaining current and [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) in the circuit ($I_R$, $U_P$, $U_R$), stating the law behind each line — then invent one extra “measurement” a careless student might report that your finished solution would instantly convict.

**Solution of Exercise 57.12.**

$I_R = 0.6 - 0.45 = 0.15\,\mathrm{A}$ ([junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law). $U_R = U_Q
= 2.3\,\mathrm{V}$ (parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law). $U_P = 6 - 2.3 =
3.7\,\mathrm{V}$ (series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law). A convictable extra: “$I_R = 0.2\,\mathrm{A}$” — the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) books would show $0.45 + 0.2 = 0.65 \neq 0.6$; or “$U_R = 2.6\,\mathrm{V}$” — parallel [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) may not disagree.

## 57.5 Problem: The Sealed Mystery Boxes

**Problem 57.1.**

Weekend problem — the electronics club’s sealed-box challenge; four laws against four mysteries; the grand audit

The club seals simple circuits into boxes, leaving only meter sockets. Teams must deduce the insides — constitution in hand. The bench [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) is $6\,\mathrm{V}$ throughout.

**Part I — Box A: two [terminals](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery), two lamps.** The label admits: two identical lamps and nothing else, either [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) or [in parallel](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel).

1. Wired to the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) , box A draws $0.2\,\mathrm{A}$ ; a single such lamp alone on $6\,\mathrm{V}$ is known to draw about $0.4\,\mathrm{A}$ . Series or parallel? Argue with a current law.
2. Predict what the box would draw in the *other* arrangement.
3. What [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) does each inner lamp receive in the actual arrangement, by which [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law?
4. In which arrangement would the hidden lamps shine at full nominal brightness — and why do they visibly not, through the box’s peephole?

**Part II — Box B: the three-lamp puzzle.** Box B admits: one lamp (X) [in series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) with a parallel pair (Y, Z — not necessarily identical).

5. The team [measures](https://one-course.com/books/physics/1/en/chapter/37-measurement-in-science-units-and-instruments#def-g6-measurement-in-science-measuring) $I_{\text{battery}} = 0.5\,\mathrm{A}$ and, at Y’s own test socket, $I_Y = 0.35\,\mathrm{A}$ . Find $I_Z$ and $I_X$ , citing laws.
6. A [voltmeter](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltmeter) across X: $2.8\,\mathrm{V}$ . Find the parallel pair’s [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) — and each of $U_Y$ , $U_Z$ .
7. Y and Z carry different currents at the same [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) . What does this reveal about the two lamps, in the language of [Example 57.5](#ex-g8-circuit-laws-unequal) ?
8. The team unscrews Y through a hatch. Predict the new $I_{\text{battery}}$ direction of change (rise, fall, or hold), reasoning from roads and their opposition.

**Part III — Box C: the cheat.**

9. Box C’s builder reports: “ [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) $6\,\mathrm{V}$ ; two series lamps inside at $2.5\,\mathrm{V}$ and $4.1\,\mathrm{V}$ ; [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) inside receiving $0.3\,\mathrm{A}$ and sending $0.2\,\mathrm{A}$ plus $0.15\,\mathrm{A}$ .” Convict the report twice, one law per conviction.
10. One of the two reported [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) readings is later found honest. If it is the $2.5\,\mathrm{V}$ , what must its partner truly be?
11. The builder confesses the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) numbers were “rounded generously”. If the two [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) readings $0.2\,\mathrm{A}$ and $0.15\,\mathrm{A}$ are honest, what must the main road truly carry?
12. Why do the four laws make sealed-box cheating a losing game? One sentence.

**Part IV — The constitution.**

13. Write the club’s charter: the four laws in four lines, each with the everyday [image](https://one-course.com/books/physics/1/en/chapter/32-mirrors-and-reflection#prop-g5-mirrors-reflection-image) that carries it (roads, tollbooths, waterfalls, forks).

**Solution of Problem 57.1.**

**1.** Series. [In series](https://one-course.com/books/physics/1/en/chapter/23-series-circuits#def-g4-series-circuits-series) the two lamps oppose the march one after the other: the loop’s current falls well below a single lamp’s $0.4\,\mathrm{A}$ — the measured $0.2\,\mathrm{A}$ fits. (Parallel would *add* two full [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel).) **2.** Parallel: two [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) of about $0.4\,\mathrm{A}$ each — roughly $0.8\,\mathrm{A}$ from the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery). **3.** The series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law splits the $6\,\mathrm{V}$ between two identical lamps: about $3\,\mathrm{V}$ each. **4.** Parallel — each lamp would span the full $6\,\mathrm{V}$. Through the peephole the series pair glows dim: each is underfed at half its usual push. **5.** $I_Z = 0.5 - 0.35 = 0.15\,\mathrm{A}$ ([junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law); $I_X = 0.5\,\mathrm{A}$ — X sits on the undivided road (series current law). **6.** $U_{\text{pair}} = 6 - 2.8 = 3.2\,\mathrm{V}$ (loop law); $U_Y = U_Z = 3.2\,\mathrm{V}$ (parallel law). **7.** Same push, different marches: the lamps oppose the current unequally — Z, carrying less at the same [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage), is the stronger opposer (the harder road). **8.** Fall: losing Y’s [branch](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) removes a road; the remaining Z-road opposes more than the pair did together, so the [battery](https://one-course.com/books/physics/1/en/chapter/16-a-first-electric-circuit#def-g3-first-electric-circuit-battery) drives less total current. **9.** Conviction one, loop law: $2.5 + 4.1 =
6.6\,\mathrm{V}$ spent from a $6\,\mathrm{V}$ purse. Conviction two, [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law: in $0.3\,\mathrm{A}$, out $0.2 + 0.15 =
0.35\,\mathrm{A}$. **10.** $6 - 2.5 = 3.5\,\mathrm{V}$. **11.** $0.2 + 0.15 = 0.35\,\mathrm{A}$. **12.** Because the four laws over-determine every honest circuit: each reading must agree with the others’ sums, so one invented number must lie to at least one law — and is convicted by arithmetic alone. **13.** For example: “One road, one march — the series current law. At every fork the marches balance, in equals out — the [junction](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) law. Around any loop, the [battery’s](https://one-course.com/books/physics/1/en/chapter/48-short-circuits-and-electrical-safety#def-g7-short-circuits-safety-battery) waterfall is spent to the last drop by the workers — the series [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law. And [branches](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) hanging between the same two [junctions](https://one-course.com/books/physics/1/en/chapter/30-series-and-parallel-circuits#def-g5-series-parallel-circuits-parallel) drink from the same height — the parallel [voltage](https://one-course.com/books/physics/1/en/chapter/56-voltage-and-the-voltmeter#def-g8-voltage-voltmeter-voltage) law.”