Chapter 43: Series and Parallel Circuits

43.1 Why Circuit Arrangement Matters

A circuit is not just a list of components — it is a system in which the arrangement of those components determines how current flows and how voltage distributes. The same three resistors wired in series, in parallel, or in a combination behave as completely different circuits. The physical laws governing each arrangement follow directly from two fundamental principles: charge cannot accumulate at a junction (conservation of charge, giving Kirchhoff's Current Law), and energy must be conserved around any closed loop (conservation of energy, giving Kirchhoff's Voltage Law). Recognising the arrangement — series, parallel, or mixed — is always the first step in circuit analysis. Once you identify the configuration, a small set of rules gives you all the currents and voltages.

Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.2 Review of Current, Voltage, and Resistance

Before proceeding, confirm these definitions from Chapter 42, because they are applied constantly in what follows. Quantity Symbol Unit Key Meaning Current I Ampere (A) Rate of charge flow; the same through any series-connected components Voltage V Volt (V) Energy per unit charge; always measured across two points Resistance R

Ohm (Ω)

Opposition to current; V = IR for

ohmic components Power P Watt (W)

P = IV = I²R = V²/R; rate of

energy dissipation

Table 43.1

43.3 Series Circuits

Components are connected in series when they are joined end-to-end so that the only path for current is through each component in succession. There is exactly one current loop; no branching occurs. If any component in a series circuit is removed or fails open -circuit, the entire circuit breaks and no current flows anywhere. The defining characteristic of a series circuit is that there is only one path for current. Because charge cannot accumulate at any point, the same current must flow through every component. This single-path property is the key to all series circuit rules.

same current. Remove any one component and the circuit is broken. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.4 Current in Series Circuits

In a series circuit, the current is the same through every component. This follows directly from conservation of charge: charge does not accumulate at any junction, and since there are no junctions in a series circuit, the current that enters one component must exit the other side and immediately enter the next.

I₁ = I₂ = I₃ = … = I

(series)

Three resistors (100 Ω, 220 Ω, 330 Ω) are connected in

series to a 12 V battery. A current meter in any part of the circuit reads the same value.

If I = 18.5 mA, then I through each resistor = 18.5 mA. (The calculation of I comes

from the equivalent resistance, covered in section 43.6.)

43.5 Voltage in Series Circuits

In a series circuit, the total voltage supplied by the source is distributed among the components. Each component has a voltage drop (also called potential difference)

across it, determined by Ohm's Law: V = IR. The sum of all these voltage drops equals

the source voltage.

V_source = V₁ + V₂ + V₃ + …

Each resistor "uses up" a portion of the available voltage proportional to its resistance. A larger resistor has a larger voltage drop across it (for the same current). The voltage drops add up to the full supply voltage — none is lost and none is created.

A 12 V battery drives a series circuit with R₁ = 100

Ω, R₂ = 200 Ω. Total R = 300 Ω. I = 12/300 = 0.040 A. V₁ = IR₁ = 0.040 × 100 = 4.0 V.

V₂ = IR₂ = 0.040 × 200 = 8.0 V. Check: 4.0 + 8.0 = 12 V = V_source. ✓The larger

resistor has the larger share of the voltage. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.6 Equivalent Resistance in Series

A set of series resistors can be replaced by a single equivalent resistor (R_eq) that

draws the same current from the source and has the same voltage across it. The equivalent resistance of series resistors is simply the sum of their individual resistances.

R_eq = R₁ + R₂ + R₃ + …

Physically, connecting resistors in series is like extending the length of a wire — each resistor adds more length of obstructing material. More length means more resistance

(recall R = ρL/A).

R_eq = 47 + 100 + 150 + 220 = 517 Ω. If the source voltage is 24 V: I = 24/517 =

0.0464 A = 46.4 mA. This single current flows through all four resistors.

Check: R_eq for series must always be larger than the largest individual

resistor. If your calculated R_eq is smaller than any single resistor in the group, you

have made an error.

43.7 Parallel Circuits

Components are connected in parallel when both terminals of each component are connected to the same two nodes. This creates multiple paths for current. Each component has a direct connection to both the positive and negative terminals of the source (or the same two nodes in a more complex circuit). Removing one parallel component leaves the others unaffected — they still have their direct connections and current still flows through them. The defining characteristic of a parallel circuit is that all components share the same two endpoints. As a result, they all experience the same voltage across their terminals. If any component is removed, the others continue operating at the same voltage.

A parallel circuit has multiple paths for current. All components share the same voltage. Remove any one component and the others are unaffected. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.8 Voltage in Parallel Circuits

In a parallel circuit, every component has the same voltage across it, equal to the voltage of the source (assuming ideal connecting wires with negligible resistance).

V₁ = V₂ = V₃ = … = V_source

(parallel) This is why household devices are wired in parallel: each device receives the full mains voltage (120 V or 230 V) regardless of what other devices are connected. A lamp, a television, and a heater plugged into the same circuit each receive the same voltage and operate independently.

43.9 Current in Parallel Circuits

In a parallel circuit, the total current from the source splits among the parallel branches.

Each branch carries a current determined by Ohm's Law and its own resistance: I = V/R.

The total current from the source is the sum of the individual branch currents.

I_total = I₁ + I₂ + I₃ + …

A lower-resistance branch carries more current (because I = V/R and V is the same for

all branches). Adding more parallel branches provides more paths for current, so the total current increases and the effective resistance decreases.

Two resistors, R₁ = 60 Ω and R₂ = 120 Ω,

are connected in parallel across a 12 V source. I₁ = 12/60 = 0.20 A. I₂ = 12/120 = 0.10

A. I_total = 0.20 + 0.10 = 0.30 A. Note: R₁ carries twice the current of R₂ because it

has half the resistance, yet both are connected to the same 12 V. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.10 Equivalent Resistance in Parallel

Parallel resistors can be replaced by a single equivalent resistor (R_eq) that draws the

same total current from the source. The reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances.

1/R_eq = 1/R₁ + 1/R₂ + 1/R₃ + …

Adding more resistors in parallel always decreases R_eq. Physically, each new parallel

branch provides an additional path for current — like adding more lanes to a highway, which increases total flow and decreases effective resistance.

R₁ = 60 Ω, R₂ = 120 Ω in parallel.

R_eq = (60 × 120)/(60 + 120) = 7200/180 = 40 Ω. Check: 40 Ω is less than both 60 Ω

and 120 Ω. ✓Source current = 12/40 = 0.30 A, confirming the branch current sum

from section 43.9.

R₁ = 30 Ω, R₂ = 60 Ω, R₃ = 90 Ω

in parallel. 1/R_eq = 1/30 + 1/60 + 1/90 = 6/180 + 3/180 + 2/180 = 11/180. R_eq =

180/11

16.4 Ω. Substantially less than 30 Ω (the smallest).

Check: R_eq for parallel must always be smaller than the smallest

individual resistor. For two equal resistors in parallel, R_eq = R/2. For n equal resistors

in parallel, R_eq = R/n.

Do not add resistances directly when resistors are in parallel.

1/R_eq = 1/R₁ + 1/R₂, so R_eq ≠R₁ + R₂. This is the most common parallel circuit

error. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.11 Combined Series-Parallel Circuits

Most real circuits are neither purely series nor purely parallel. They contain groups of series-connected components and groups of parallel-connected components combined together. The strategy is to reduce the circuit step by step, simplifying the most nested sub-circuits first, until a single equivalent resistance remains.

Step 1: Identify:

Find groups of resistors that are clearly in series or clearly in parallel with each other. Start with the most nested (innermost) group.

Step 2: Simplify:

Replace each identified group with its equivalent resistance using the appropriate formula.

Step 3: Repeat:

Redraw the simplified circuit. Identify the next group to simplify. Repeat until one equivalent resistor remains.

Step 4: Find I and V: With R_eq and the source voltage known, find the total current:

I = V/R_eq. Then work backwards through the circuit, applying Kirchhoff's rules at each

junction and Ohm's Law across each component to find individual currents and voltages.

Three resistors: R₁ = 100 Ω in series with a parallel

combination of R₂ = 150 Ω and R₃ = 300 Ω. Connected to a 24 V source. Step 1: Find

R₂″R₃ = (150 × 300)/(150 + 300) = 45 000/450 = 100 Ω. Step 2: R_eq = R₁ +

R_parallel = 100 + 100 = 200 Ω. Step 3: I_total = 24/200 = 0.120 A. Voltage across R₁:

V₁ = 0.120 × 100 = 12.0 V. Voltage across parallel group: 24

12 = 12.0 V (or: V =

I_total × R_parallel = 0.120 × 100 = 12 V). I₂ = 12/150 = 0.080 A; I₃ = 12/300 = 0.040

A. Check: 0.080 + 0.040 = 0.120 A = I_total. ✓

Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.12 Circuit Analysis Strategy

Approaching a complex circuit systematically prevents errors. Apply the following decision process. Question If yes → Applicable Rule Are all components connected end-to-end (one path)? Series

I same everywhere; V adds; R_eq =

sum of R Do all components share the same two nodes? Parallel

V same everywhere; I adds; 1/R_eq

= sum of 1/R Is there a mix? Combined Reduce innermost sub-circuits first; redraw and repeat

Table 43.2

After finding all currents and voltages, always verify using two checks: (1) the sum of all voltage drops across components in a loop must equal the source voltage (Kirchhoff 's Voltage Law), and (2) the current entering any junction must equal the current leaving it (Kirchhoff's Current Law). If either check fails, an error exists somewhere. Series Parallel Current

Same through all: I₁ = I₂ = …

Splits: I_T = I₁ + I₂ + …

Voltage

Splits: V_T = V₁ + V₂ + …

Same across all: V₁ = V₂ = …

Equivalent R

R_eq = R₁ + R₂ + … (larger than

any R)

1/R_eq = Σ 1/R (smaller than

any R) If one fails open All components stop Others unaffected Application Voltage dividers, fuses, switches Household wiring, parallel loads

Table 43.3

Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.13 Practical Circuit Design Considerations

Real circuit design uses series and parallel arrangements for specific purposes. Understanding the physics behind each motivates the engineering decisions. Voltage Dividers (Series) Two resistors in series with a power supply form a voltage divider. The output voltage is taken from the junction between the two resistors. If R₁ and R₂ are in series with

supply voltage V_s, the voltage across R₂ is: V_out = V_s × R₂/(R₁ + R₂). This

relationship allows precise output voltages to be produced from a fixed supply. Fuses and Circuit Breakers (Series) Fuses and circuit breakers are always placed in series with the circuit they protect. When current exceeds the fuse rating, the fuse element melts (or the breaker trips), opening the circuit. Because fuses are in series with the load, opening the fuse stops current through the entire load. A fuse in parallel would not work — it would simply provide a low-resistance bypass and bypass current would increase when the fuse blew. Household Wiring (Parallel) All household outlets are wired in parallel from the main distribution panel. This ensures each appliance receives the full mains voltage (120 V or 230 V) regardless of how many others are plugged in. It also allows individual control: unplugging one device does not affect others. The tradeoff is that adding more parallel loads increases total current from the source, which is why circuit breakers have current ratings.

Thomas Edison's original DC power distribution (1882) and Nikola Tesla's AC distribution (1888–1893) both used parallel connections to supply customers with a standard voltage. The "War of Currents" was not about series vs. parallel but about DC vs. AC — both systems connected loads in parallel to maintain a constant voltage for each customer. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

43.14 Common Mistakes in Circuit Analysis

•

Adding resistances in parallel instead of using 1/R_eq = Σ 1/R. Parallel

resistances combine as reciprocals, not directly. The result must be smaller than the smallest individual resistor. • Assuming current is the same everywhere in a parallel circuit. Current splits in parallel circuits. Only voltage is the same for all parallel branches. Current equality applies only in series circuits. • Assuming voltage splits in a parallel circuit. All parallel branches share the same voltage. Only in series circuits does voltage split across components. • Forgetting to take the reciprocal at the end of the parallel formula. After

computing 1/R₁ + 1/R₂, many students forget to invert the sum. R_eq = 1 / (1/R₁

+ 1/R₂), not 1/R₁ + 1/R₂ itself. • Misidentifying series vs. parallel in a combined circuit. Two components are in series only if the same current must pass through both. Two components are in parallel only if both terminals of one are connected to both terminals of the other. Redraw the circuit if necessary to clarify the topology. • Not checking the answer with Kirchhoff's laws. Voltage drops around any loop must sum to zero (when accounting for the source). Current into any junction must equal current out. These checks catch arithmetic errors before they propagate.

When working through a combined circuit, always redraw the circuit after each simplification step. Drawing it out prevents mistakes that come from trying to track the simplification mentally. Label each equivalent resistance clearly. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

Chapter Summary

Series and parallel are the two fundamental circuit arrangements. All multi-component circuits are built from combinations of these two configurations. •

Series: one path; I same everywhere; V splits (V_T = ΣV); R_eq = ΣR (larger than

any individual R). •

Parallel: multiple paths; V same everywhere; I splits (I_T = ΣI); 1/R_eq = Σ(1/R)

(smaller than any individual R). • Kirchhoff's Voltage Law: sum of voltage drops around any closed loop equals the source voltage. • Kirchhoff's Current Law: sum of currents entering any junction equals sum leaving. • Combined circuits: reduce innermost sub-circuits first; work outward to a single

R_eq; then work backwards to find individual V and I.

• Applications: series for voltage dividers, fuses, and switches; parallel for household wiring and independent loads. • Always verify answers using KVL and KCL checks. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

Key Terms

series circuit

Circuit arrangement with a single current path; all components carry the

same current; voltages add; R_eq = sum of individual R values.

parallel circuit

Circuit arrangement where both terminals of each component share

the same two nodes; all components have the same voltage; currents add; 1/R_eq =

sum of 1/R.

equivalent resistance (R_eq)

A single resistance that produces the same total current

from the source as the entire combination it replaces.

Kirchhoff's Voltage Law (KVL)

The sum of all voltage drops around any closed loop

equals the source voltage (equivalently, net voltage change around any closed loop is zero).

Kirchhoff's Current Law (KCL)

The sum of currents entering any junction equals the

sum of currents leaving that junction; a consequence of conservation of charge.

voltage divider

Two resistors in series used to produce an output voltage that is a

fraction of the supply voltage

V_out = V_s × R₂/(R₁ + R₂).

node (junction)

A point in a circuit where two or more components connect. In

parallel circuits, all components share the same two nodes.

branch

A single path through a component or group of components in a parallel

circuit; each branch carries its own current.

open circuit

A break in a circuit that prevents current flow in that path. In series: all

current stops. In parallel

only the open branch stops.

short circuit

A low-resistance path that allows very large current to flow, bypassing

the intended load. Dangerous

can cause overheating, fire, or equipment damage.

Physics

Foundations of Physical Science

Chapter 43 — Series and Parallel Circuits

Concept Review Questions

Answer in complete sentences. No calculations required unless stated.

1. Explain why adding more resistors in series always increases the total resistance,

but adding more resistors in parallel always decreases the total resistance. Use the physics of charge flow to justify each.

2. In a series circuit, the largest resistor has the largest voltage drop. Explain why

this follows from Ohm's Law and the series current rule.

3. Household outlets are wired in parallel, not in series. Explain two practical

advantages of parallel wiring for home circuits.

4. A string of old Christmas lights was wired in series. When one bulb burned out,

all the lights went dark. Explain why this happened in series but would not have happened if the lights were wired in parallel.

5. State Kirchhoff's Current Law and Kirchhoff's Voltage Law in your own words.

Which conservation law underlies each?

6. Two resistors are connected in parallel. One has resistance R; the other also has

resistance R. What is their equivalent resistance? What if three identical R resistors are connected in parallel? Generalise.

7. A fuse is always placed in series with the circuit it protects, never in parallel.

Explain why a parallel fuse would not protect the circuit.

8. In a combined series-parallel circuit, why do you always simplify the most deeply

nested sub-circuit first? Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

Series Circuit Problems

9. Three resistors (R₁ = 22 Ω, R₂ = 47 Ω, R₃ = 100 Ω) are connected in series to a

9.0 V battery. Find: (a) the equivalent resistance, (b) the current, (c) the voltage across each resistor, (d) the power dissipated in each resistor.

10. A 6.0 V battery drives a series circuit containing a 150 Ω resistor and an

unknown resistor R_x. A current of 25 mA flows. Find R_x.

11. A voltage divider consists of R₁ = 1200 Ω and R₂ = 800 Ω in series with a 15 V

source. Find the voltage across R₂ (the output voltage).

12. Four identical 68 Ω resistors are in series with a 24 V supply. Find the current,

the voltage across each, and the total power dissipated.

13. Three lamps are connected in series to a 12 V battery. Lamp A has resistance 4.0

Ω; Lamp B has resistance 6.0 Ω; Lamp C has resistance 2.0 Ω. Find the current

and the voltage across each lamp. Which lamp is brightest?

Parallel Circuit Problems

14. Three resistors (R₁ = 60 Ω, R₂ = 120 Ω, R₃ = 40 Ω) are connected in parallel

across a 24 V source. Find: (a) R_eq, (b) branch currents, (c) total current, (d)

power dissipated in each branch.

15. Two resistors in parallel have an equivalent resistance of 24 Ω. One resistor has

resistance 40 Ω. Find the other.

16. A 120 V household circuit has a 60 W lamp, a 100 W lamp, and a 40 W lamp all

connected in parallel. Find: (a) the resistance of each lamp, (b) the current through each, (c) the total current from the source, (d) the total power.

17. How many 60 Ω resistors connected in parallel are needed to achieve an

equivalent resistance of 12 Ω?

18. A 12 V battery has internal resistance 0.5 Ω. Two external resistors (R₁ = 10 Ω, R₂

= 20 Ω) are connected in parallel. Find the terminal voltage of the battery (hint:

the internal resistance is in series with the parallel combination). Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

Equivalent Resistance Exercises

19. Find the equivalent resistance of: (a) 10 Ω, 20 Ω, 30 Ω all in series. (b) 10 Ω, 20

Ω, 30 Ω all in parallel. (c) 10 Ω in series with the parallel combination of 20 Ω

and 30 Ω.

20. Three resistors are in parallel: R₁, R₂ = 2R₁, R₃ = 3R₁. Express R_eq in terms of R₁.
21. A ladder of five 100 Ω resistors is arranged as follows: two in series forming one

branch, one alone forming a second branch, and two in series forming a third

branch, all three branches in parallel. Find R_eq.

22. What value resistor must be added in parallel with a 120 Ω resistor to produce

an equivalent resistance of 80 Ω?

Mixed Circuit Analysis

23. A circuit has R₁ = 50 Ω in series with the parallel combination of R₂ = 100 Ω and

R₃ = 100 Ω. The source voltage is 30 V. Find (a) R_eq, (b) total current, (c)

voltage across R₁, (d) voltage across the parallel pair, (e) current through each of R₂ and R₃.

24. A circuit has a parallel combination of R₁ = 40 Ω and R₂ = 60 Ω, in series with

R₃ = 10 Ω, connected to a 24 V battery. Find all branch currents, all voltage

drops, and verify with KVL.

25. Three resistors are connected: R₁ and R₂ are in parallel; this parallel pair is in

series with R₃; R₃ is in series with R₄; the whole assembly connects to a 48 V

source. Given R₁ = 60 Ω, R₂ = 40 Ω, R₃ = 5 Ω, R₄ = 10 Ω, find: (a) R_eq, (b) total

current, (c) all individual currents and voltages.

26. A circuit is powered by a 12 V battery (assume ideal). It consists of two branches

in parallel: Branch 1 contains two resistors in series (R₁ = 30 Ω, R₂ = 50 Ω);

Branch 2 contains one resistor (R₃ = 40 Ω). Find all currents and voltages. Then

find the total power delivered by the battery and confirm it equals the total power dissipated in all resistors. Physics: Foundations of Physical Science Chapter 43 — Series and Parallel Circuits

Chapter 43 Checkpoint

Before moving on to Chapter 44, confirm you can do each of the following. I can… See Section Identify whether resistors are in series, parallel, or a combination

43.3 / 43.7 / 43.11

Apply the series current rule: same current through all series components 43.4

Apply the series voltage rule: V_T = ΣV; use KVL to verify

43.5

Calculate equivalent resistance in series: R_eq = ΣR

43.6 Apply the parallel voltage rule: same voltage across all parallel components 43.8

Apply the parallel current rule: I_T = ΣI; use KCL to verify

43.9

Calculate equivalent resistance in parallel: 1/R_eq = Σ(1/R)

43.10 Analyse combined series-parallel circuits by stepwise reduction 43.11–43.12 Explain why household wiring uses parallel connections 43.13 Identify and avoid common circuit analysis mistakes 43.14