Lesson 3 of 5Electricity and Magnetism

Unit 5 · Lesson 3 of 5

Series and Parallel Circuits

The two ways to connect components, how resistances combine in each, and the current and voltage rules that follow.

4 min read10 practice questionsInteractiveAsk AI about this →

Key things to remember

1Series: $R_{T} = R_{1} + R_{2} + \dots$ — the total exceeds any individual resistor.
2Parallel: $\frac{1}{R _{T}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots$ — the total is less than the smallest resistor.
3Remember the final reciprocal in a parallel calculation.
4Series: current is the same everywhere; voltages add to the supply.
5Parallel: voltage is the same across every branch; currents add to the total.
6In series the largest resistor takes the largest voltage; in parallel the smallest resistor takes the largest current.
7Two equal resistors in parallel give half of one. For two unequal ones, $R_{T} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$ .
8Houses are wired in parallel so every outlet gets full voltage and devices work independently.

There are exactly two ways to connect two components. In series they sit one after another on a single path, so all the current must pass through both. In parallel they sit on separate branches, so the current divides between them. Everything else follows from that one distinction.

Fig 1One path or several. The rules for current and voltage are mirror images of each other.

Series circuits

R_{T, ser i es} = R_{1} + R_{2} + \dots

Resistances simply add — the total is always larger than any individual resistor.

Current is the same everywhere. There is only one path, so every charge passes through every component: $I_{T} = I_{1} = I_{2}$ .
Voltages add to the supply. Each resistor takes a share, and the shares sum to the source voltage: $V_{T} = V_{1} + V_{2}$ .
The largest resistor takes the largest share of the voltage, in direct proportion to its resistance.
Adding a resistor increases the total resistance, so the current falls everywhere.
One break stops everything — the single path is broken.

Parallel circuits

\frac{1}{R _{T}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots

Reciprocals add. The total is always smaller than the smallest individual resistor.

Do not forget the final reciprocal

The equation gives you

\frac{1}{R _{T}}

, not

R_{T}

. Flipping it at the end is the single most common error in this topic. The check is instant: if your total resistance is larger than the smallest branch, you skipped the last step.

Voltage is the same across every branch. Each branch connects directly to both terminals: $V_{T} = V_{1} = V_{2}$ .
Currents add to the total. The supply current splits and recombines: $I_{T} = I_{1} + I_{2}$ .
The smallest resistor takes the largest current, since $I = \frac{V}{R}$ with $V$ fixed.
Adding a branch decreases total resistance, so the total current rises.
One break affects only that branch — everything else keeps working.

ASeries

$R_{T} = R_{1} + R_{2}$ — total goes up
Current the same everywhere
Voltages add to the supply
One break stops everything
Used in fuses and switches, which must interrupt the whole circuit

BParallel

$\frac{1}{R _{T}} = \frac{1}{R _{1}} + \frac{1}{R _{2}}$ — total goes down
Voltage the same across every branch
Currents add to the total
One break affects only that branch
Used for household wiring and vehicle lights

Why houses are wired in parallel

Every outlet gets the full

120 V

, appliances can be switched independently, and one failed bulb does not darken the house. The trade-off is that each added appliance lowers the total resistance and raises the total current — which is exactly what a circuit breaker is there to limit.

Two special cases worth memorizing

Two equal resistors in parallel give exactly half of one of them. Two unequal resistors in parallel give

R_{T} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}

— the product over the sum — which avoids the reciprocal mistake entirely. Both are faster than the general formula when only two resistors are involved.

Loading interactive…

Build a series or parallel network and read the total resistance, branch currents and power for each resistor.

Working through a circuit

Find the total resistance using the appropriate combination rule.
Find the total current from the supply voltage: $I_{T} = \frac{V _{T}}{R _{T}}$ .
Work back to individual components. In series, the current is known everywhere, so use $V = I R$ on each resistor. In parallel, the voltage is known everywhere, so use $I = \frac{V}{R}$ on each branch.
Check with the rules. Series voltages must sum to the supply; parallel currents must sum to the total. If they do not, something is wrong.

Short circuits

A short circuit is a near-zero-resistance path across the supply. Ohm's law then predicts an enormous current, which is why a short blows fuses, melts insulation and starts fires. Fuses and breakers exist precisely to interrupt the circuit before that current does damage.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Three resistors of

10 Ω

15 Ω

and

25 Ω

are connected in series to a

12 V

battery. Find the total resistance, the current, and the voltage across each resistor.

Example 2

Two resistors of

6.0 Ω

and

12 Ω

are connected in parallel across a

24 V

supply. Find the total resistance, the current in each branch, and the total current.

Example 3

Three

30 Ω

resistors are connected in parallel. Find the total resistance.

Where marks get lost

Forgetting the final reciprocal in a parallel calculation, leaving the answer as $\frac{1}{R _{T}}$ .
Adding resistances in parallel as though they were in series, which always gives a total that is far too large.
Expecting the current to divide equally between unequal parallel branches. The smaller resistance takes more.
Assuming the voltage divides equally between unequal series resistors. It divides in proportion to resistance.
Believing the current is 'used up' as it passes through a series circuit. It is identical at every point.
Applying series voltage rules to a parallel circuit, or parallel current rules to a series one.
Never checking the answer. Series voltages must sum to the supply, and parallel currents must sum to the total.

Flashcards 11

Series resistance

Parallel resistance

Current in a series circuit

Voltage in a series circuit

+ 7 more

Practice

Resistors of

8.0 Ω

12 Ω

and

20 Ω

are connected in series. What is the total resistance, in ohms?

ohms

Two

20 Ω

resistors are connected in parallel. What is the total resistance, in ohms?

ohms

4.0 Ω

and a

6.0 Ω

resistor are in parallel. What is the total resistance, in ohms?

ohms

In a series circuit, what is true about the current?

In a parallel circuit, which resistor carries the largest current?

6.0 V

battery is connected to two resistors in series,

4.0 Ω

and

8.0 Ω

. What current flows, in amperes?

Adding another appliance to a household parallel circuit has what effect?

In a parallel resistance calculation, what step is most often forgotten?

Match each property to the circuit type it describes.

Current is the same everywhere

Voltage is the same across each branch

One break stops the whole circuit

Total resistance is less than the smallest resistor

Order the steps for analysing a circuit.

1Find the total current from the supply voltage
2Check that series voltages sum to the supply, or parallel currents to the total
3Work back to find the voltage or current at each component
4Find the total resistance using the appropriate combination rule

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