Why is the equivalent resistance smaller in parallel?

Each branch offers an extra path to the current: under the same voltage the total current is larger, so U/I is smaller. Two equal resistors in parallel are worth half of one.

Are the lamps of a house in series or in parallel?

In parallel: each receives the 230 V of the mains and works independently of the others. In series, switching one lamp off would switch off all the others, and each would receive only a fraction of the voltage.

Reminder sheet

Resistor networks

Bastaya

In series, resistances add up and the current is shared; in parallel, conductances add up and the voltage is shared. Every network reduces to these two rules.

In series the same current flows through every resistor: resistances add up and the voltage is shared in proportion to each (voltage divider).

In parallel all branches are at the same voltage: currents add up and the equivalent resistance is smaller than the smallest branch. A mixed network is reduced step by step. The two wires of a line are in series with the load: they cause a voltage drop 2·R_l·I.

Formulas

Whatever the network

Ohm's law on each resistorU_n = R_n · I_n
I_n = U_nR_n
R_n = U_nI_n

Quantity	Symbol	Unit
Voltage across resistor n	U_n	V
Resistor n	R_n	Ω
Current through it	I_n	A

Every resistor obeys Ohm's law with its own voltage and current, in series as in parallel.

Equivalent resistanceRéq = UI

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω
Voltage across the network	U	V
Total current	I	A

Req is the single resistor that, at the same voltage U, would carry the same current I.

Power in each resistorP_n = U_n · I_n
P_n = R_n · I_n²
P_n = U_n²R_n

Quantity	Symbol	Unit
Power dissipated by resistor n	P_n	W

Power balanceP = P₁ + P₂ + ...
P = U · I

Quantity	Symbol	Unit
Total power	P	W

Powers always add up, in series, in parallel or in a mixed network.

In series

Equivalent resistanceRéq = R₁ + R₂ + ...

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω

Common and split quantitiesI = I₁ = I₂ = ...
U = U₁ + U₂ + ...

Quantity	Symbol	Unit
Current (common)	I	A
Total voltage (split)	U	V

In series the current I is common to every resistor; the voltage U is split between them.

Voltage across each resistorU = Réq · I
U₁ = R₁ · I
U_n = R_n · I

Quantity	Symbol	Unit
Voltage across resistor n	U_n	V
Resistor n	R_n	Ω
Current	I	A

CurrentI = URéq
I₁ = U₁R₁
I_n = U_nR_n

Quantity	Symbol	Unit
Current	I	A
Total voltage	U	V
Equivalent resistance	Réq	Ω

Each resistanceRéq = UI
R₁ = U₁I
R_n = U_nI

Quantity	Symbol	Unit
Resistor n	R_n	Ω
Voltage across resistor n	U_n	V
Current	I	A

Voltage dividerU₁ = U · R₁R₁ + R₂

Quantity	Symbol	Unit
Voltage on R₁	U₁	V
Total voltage	U	V

In parallel

Equivalent resistance1Réq = 1R₁ + 1R₂ + ...

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω

Two resistors: Req = R₁·R₂ / (R₁ + R₂).

Common and split quantitiesU = U₁ = U₂ = ...
I = I₁ + I₂ + ...

Quantity	Symbol	Unit
Voltage (common)	U	V
Total current (split)	I	A

In parallel the voltage U is common to every branch; the current I is split between them.

Current in each branchI = URéq
I₁ = UR₁
I_n = UR_n

Quantity	Symbol	Unit
Current in branch n	I_n	A
Voltage (common)	U	V
Resistance of branch n	R_n	Ω

The branch with the smallest resistance carries the largest current.

VoltageU = Réq · I
U = R₁ · I₁ = R₂ · I₂ = ... = R_n · I_n

Quantity	Symbol	Unit
Voltage (common)	U	V
Total current	I	A
Equivalent resistance	Réq	Ω

Resistance of each branchRéq = UI
R₁ = UI₁
R_n = UI_n

Quantity	Symbol	Unit
Resistance of branch n	R_n	Ω
Voltage (common)	U	V
Current in branch n	I_n	A

Current dividerI₁ = I · R₂R₁ + R₂

Quantity	Symbol	Unit
Current in R₁	I₁	A
Total current	I	A

Two branches: the current splits inversely to the resistances (I₁ gets R₂'s share).

n identical resistorsRéq = Rn

Quantity	Symbol	Unit
Equivalent resistance	Réq	Ω
Resistance of each branch	R	Ω
Number of resistors	n	—

Mixed network

Step-by-step reductionR₂₃ = R₂ · R₃R₂ + R₃
Réq = R₁ + R₂₃

Quantity	Symbol	Unit
Equivalent resistance of R₂ ∥ R₃	R₂₃	Ω
Total equivalent resistance	Réq	Ω

Example: R₁ in series with R₂ ∥ R₃. Replace each series or parallel group by its equivalent resistance until a single resistor remains.

Other case: parallel then seriesR₂₃ = R₂ + R₃
Réq = R₁ · R₂₃R₁ + R₂₃

Quantity	Symbol	Unit
Equivalent resistance of R₂ + R₃	R₂₃	Ω
Total equivalent resistance	Réq	Ω

R₁ in parallel with the R₂ + R₃ branch: reduce the series branch first, then the parallel.

Total currentI = URéq

Quantity	Symbol	Unit
Current delivered by the generator	I	A
Generator voltage	U	V

Back down to each resistorU₁ = R₁ · I
U₂₃ = R₂₃ · I
I₂ = U₂₃R₂
I₃ = U₂₃R₃

Quantity	Symbol	Unit
Voltage on R₁	U₁	V
Voltage shared by R₂ and R₃	U₂₃	V
Current in R₂	I₂	A

Walk back the other way: a series group gets the common current (U = R · I), a parallel group gets the common voltage (I = U / R). Check: U₁ + U₂₃ = U and I₂ + I₃ = I.

Power line

Line voltage dropΔU = 2 · R_l · I

Quantity	Symbol	Unit
Voltage drop	ΔU	V
Resistance of one wire	R_l	Ω
Current	I	A

Worked example

R₁ = 100 Ω is in series with the group R₂ = 220 Ω ∥ R₃ = 330 Ω, all under U = 12 V. Find the equivalent resistance, the total current, then the voltage and current of each resistor.

Parallel first: R₂₃ = 220 × 330 / (220 + 330) = 132 Ω. Then series: Req = 100 + 132 = 232 Ω.

I = U / Req = 12 / 232 = 51.7 mA.

U₁ = R₁ · I = 100 × 0.0517 = 5.17 V; U₂₃ = 132 × 0.0517 = 6.83 V (check: 5.17 + 6.83 = 12 V).

I₂ = U₂₃ / R₂ = 6.83 / 220 = 31.0 mA; I₃ = 6.83 / 330 = 20.7 mA (check: 31.0 + 20.7 = 51.7 mA).

Common mistakes

Adding parallel resistors as if they were in series: in parallel, the reciprocals add up, and Req is smaller than the smallest branch.
Forgetting to invert back: 1/Req = 0.00758 is not the answer, Req = 1 / 0.00758 = 132 Ω.
Applying the voltage divider to parallel resistors: they share the same voltage, it is the current that splits.

IEC symbols

Resistor

Mini circuit

Open in the sandbox

Frequently asked questions

Why is the equivalent resistance smaller in parallel?: Each branch offers an extra path to the current: under the same voltage the total current is larger, so U/I is smaller. Two equal resistors in parallel are worth half of one.
Are the lamps of a house in series or in parallel?: In parallel: each receives the 230 V of the mains and works independently of the others. In series, switching one lamp off would switch off all the others, and each would receive only a fraction of the voltage.

71 exercises in this topic

Two resistors in series Build, Analyse · level 2
Voltage sharing Analyse · level 2
Equivalent resistance Build, Analyse · level 3
Two resistors in parallel Build, Analyse · level 2
Branch current Analyse · level 2
Aim for a parallel resistance Build · level 3
Mixed circuit Build, Analyse · level 3
Short at the source Build · level 2
Wrong value Build, Analyse · level 3
Two faults Build · level 3
Tricky mixed circuit Build · level 4
Current in series Analyse · level 2
Voltage on R1 (series) Analyse · level 2
Voltage on R2 (series) Analyse · level 2
Req of three resistors Analyse · level 2
Current in three resistors Analyse · level 2
Sum of voltages Analyse · level 2
Reproduce: two in series Build, Analyse · level 3
Total current in parallel Analyse · level 2
Current in a branch Analyse · level 2
Req in parallel Analyse · level 2
Voltage on a branch Analyse · level 2
Three in parallel Analyse · level 2
Reproduce: two in parallel Build, Analyse · level 3
Req of a mixed circuit Analyse · level 2
Total current (mixed) Analyse · level 2
Voltage on R1 (mixed) Analyse · level 2
Current in R3 (mixed) Analyse · level 2
The hungriest branch Analyse · level 2
The largest voltage drop Analyse · level 2
Series: the current Know · level 1
Parallel: the voltage Know · level 1
Req in parallel Know · level 1
Short circuit Know · level 1
Line voltage drop Calculate · level 3
n identical resistors Calculate · level 2
Mixed network Calculate · level 4
Two-wire line Analyse · level 3
Equivalent resistance in series Calculate · level 1
Equivalent resistance in parallel Calculate · level 2
Equivalent resistance of a mixed group Calculate · level 3
Series: from partial to total voltage Calculate · level 2
Series: from total to partial voltages Calculate · level 2
Series: current from the total power Calculate · level 3
Series: unknown resistance from a partial power Calculate · level 4
Parallel: current division Calculate · level 2
Parallel: unknown resistance Calculate · level 3
Resistors given by their rating Calculate · level 3
Oven settings: in parallel Calculate · level 3
Oven settings: in series Calculate · level 3
Line voltage drop: source voltage Calculate · level 3
Line voltage drop: admissible resistance Calculate · level 4
Conductor cut into pieces put in parallel Calculate · level 3
n identical resistors Calculate · level 1
Build a series circuit and read a partial voltage Build, Analyse · level 2
Build a parallel circuit and find the total current Build, Analyse · level 2
Equivalent resistance of a network Analyse · level 3
Three identical resistors, one value to reach Build, Analyse · level 3
The missing resistor Build, Calculate · level 3
Current divider Analyse · level 3
Oven setting 1: a single resistor Build, Calculate · level 2
Oven setting 2: two resistors in parallel Build, Calculate · level 2
Oven setting 3: two resistors in series Build, Calculate · level 2
Rank the oven settings Know · level 2
Compensate the line voltage drop Build, Calculate · level 3
Cross-section of a garage line Build, Calculate · level 4
Troubleshoot a series circuit Build · level 3
Who is in parallel, who carries the most current? Analyse · level 3
Two halves of a wire in parallel Analyse · level 3
Three loads in parallel: Req Analyse · level 3
Find R₁ thanks to the wattmeter Measure, Calculate · level 4

Practise this topic

Simulated time00:00:00

Q = I·t-

W = P·t-