Series Circuits

What Is a Series Circuit?

A series circuit is a circuit in which components are connected one after another in a single continuous path.

Because there is only one path for charge to follow, the same current passes through every component in the circuit.

A simple series circuit might contain:

• a battery or power supply
• a switch
• two or more resistors or lamps
• connecting wires

The main rules for a series circuit are:

Current is the same everywhere.

Voltages across components add to the supply voltage.

Resistances add together.

These three rules allow us to analyze most simple series circuits.

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Current in a Series Circuit

Because a series circuit has only one path, charge cannot split into different branches.

Therefore:

I_total = I₁ = I₂ = I₃ = ...

If the current leaving the battery is:

2 A

then the current through every component in the series circuit is also:

2 A

The current is not used up as it moves around the circuit.

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Why Is Current the Same?

Current is the rate of flow of charge:

I = Q/t

where:

• I = current
• Q = charge
• t = time

Suppose 10 C of charge passes one point in the circuit during a certain time.

The same 10 C must pass the next component.

Charge does not normally accumulate inside a resistor or lamp.

Therefore, the rate of charge flow must be the same throughout a steady series circuit.

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Measuring Current

Current is measured using an ammeter.

An ammeter is connected in series with the component being measured.

Because the current is the same throughout a series circuit, ammeters placed at different positions should give approximately the same reading.

For example:

A₁ = 0.50 A

A₂ = 0.50 A

A₃ = 0.50 A

Small differences in a real experiment may occur because of measurement uncertainty.

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Voltage in a Series Circuit

Voltage behaves differently from current.

The supply voltage is shared between the components.

For a series circuit:

V_total = V₁ + V₂ + V₃ + ...

For example, suppose a battery provides:

12 V

and two resistors have voltage drops of:

V₁ = 4 V

V₂ = 8 V

Then:

V_total = 4 + 8

V_total = 12 V

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What Does a Voltage Drop Mean?

A voltage drop represents energy transferred per unit charge by a component.

As charge moves through a resistor, electrical energy may be transferred into:

• thermal energy
• light
• sound
• mechanical energy
• other forms

A component with a larger voltage drop transfers more energy per unit charge.

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Measuring Voltage

Voltage is measured using a voltmeter.

Unlike an ammeter, a voltmeter is connected in parallel across a component.

For example, if two resistors are connected in series, separate voltmeter readings might be:

V₁ = 3 V

V₂ = 6 V

If the power supply is 9 V:

V₁ + V₂ = V_supply

3 + 6 = 9 V

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Resistance in Series

When resistors are connected in series, their resistances add.

R_total = R₁ + R₂ + R₃ + ...

For example:

R₁ = 4 Ω

R₂ = 6 Ω

Then:

R_total = 4 + 6

R_total = 10 Ω

If a third resistor of 5 Ω is added:

R_total = 4 + 6 + 5

R_total = 15 Ω

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Why Do Series Resistances Add?

Each resistor provides additional opposition to the movement of charge.

A charge moving around the circuit must pass through:

• resistor 1
• then resistor 2
• then resistor 3

Adding another resistor therefore creates additional resistance along the only available path.

This means:

adding resistance in series always increases the total resistance.

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Equivalent Resistance

Several resistors can be replaced mathematically by a single resistor that has the same effect on the circuit.

This is called the equivalent resistance.

For resistors in series:

R_eq = R₁ + R₂ + R₃ + ...

Suppose:

R₁ = 3 Ω

R₂ = 5 Ω

R₃ = 7 Ω

Then:

R_eq = 3 + 5 + 7

R_eq = 15 Ω

The three resistors together behave like one 15 Ω resistor from the perspective of the power supply.

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Equal Resistors in Series

If several identical resistors are connected in series:

R_total = nR

where:

• n = number of resistors
• R = resistance of each resistor

For example, four 10 Ω resistors give:

R_total = 4 × 10

R_total = 40 Ω

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Combining Series Resistance with Ohm's Law

Once the equivalent resistance is known, Ohm's Law can be applied to the entire circuit.

V = IR

or:

I = V/R

Suppose:

V_supply = 12 V

R₁ = 2 Ω

R₂ = 4 Ω

First calculate total resistance:

R_total = 2 + 4

R_total = 6 Ω

Then calculate current:

I = V/R

I = 12/6

I = 2 A

Since this is a series circuit:

I₁ = I₂ = 2 A

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Finding Individual Voltage Drops

Continue with the previous circuit:

I = 2 A

R₁ = 2 Ω

R₂ = 4 Ω

For resistor 1:

V₁ = IR₁

V₁ = 2 × 2

V₁ = 4 V

For resistor 2:

V₂ = IR₂

V₂ = 2 × 4

V₂ = 8 V

Check:

V₁ + V₂ = V_supply

4 + 8 = 12 V

The calculation is consistent.

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Larger Resistance Gets a Larger Voltage Drop

In a series circuit, the same current flows through every resistor.

Since:

V = IR

a larger resistance produces a larger voltage drop.

In the previous example:

2 Ω resistor → 4 V

4 Ω resistor → 8 V

The 4 Ω resistor has twice the resistance, so it has twice the voltage drop.

Therefore, for resistors in series:

V₁/V₂ = R₁/R₂

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Voltage Division

The sharing of supply voltage between series resistors is called voltage division.

For a resistor R₁:

V₁ = V_total × R₁/R_total

Suppose:

V_total = 15 V

R₁ = 2 Ω

R₂ = 3 Ω

Total resistance:

R_total = 5 Ω

Voltage across R₁:

V₁ = 15 × 2/5

V₁ = 6 V

Voltage across R₂:

V₂ = 15 × 3/5

V₂ = 9 V

Check:

6 + 9 = 15 V

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Equal Resistors Share Voltage Equally

Suppose three identical resistors are connected in series across a 12 V supply.

Because their resistances are equal, their voltage drops are equal.

Therefore:

V_each = 12/3

V_each = 4 V

This principle is useful when analyzing circuits containing identical components.

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Worked Example: Three Resistors

A circuit contains:

V_supply = 24 V

R₁ = 2 Ω

R₂ = 4 Ω

R₃ = 6 Ω

First calculate total resistance:

R_total = 2 + 4 + 6

R_total = 12 Ω

Calculate current:

I = V/R

I = 24/12

I = 2 A

The current through every resistor is:

2 A

Now calculate the voltage drops.

V₁ = 2 × 2 = 4 V

V₂ = 2 × 4 = 8 V

V₃ = 2 × 6 = 12 V

Check:

4 + 8 + 12 = 24 V

Everything agrees with the series circuit rules.

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Predicting the Effect of Adding a Resistor

Suppose a series circuit initially contains:

V = 12 V

R = 4 Ω

Current:

I = 12/4

I = 3 A

Now add another 2 Ω resistor in series.

New total resistance:

R_total = 4 + 2

R_total = 6 Ω

New current:

I = 12/6

I = 2 A

Therefore:

adding a resistor in series increases total resistance and decreases current.

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What Happens to Lamps?

Suppose identical lamps are connected in series to a battery.

Adding another lamp:

• increases total resistance
• decreases the circuit current
• changes how the supply voltage is shared
• generally makes each identical lamp dimmer

This is one reason household lighting is not normally wired as a simple series circuit.

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What Happens If a Component Is Removed?

A series circuit has only one path.

If a component is removed and the circuit becomes open:

current stops everywhere in the circuit.

The same occurs if a lamp filament breaks.

One break interrupts the entire current path.

This is an important disadvantage of series connections in some applications.

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Adding Cells in Series

Cells can also be connected in series.

When correctly oriented in the same direction, their voltages add.

For example, two 1.5 V cells in series provide approximately:

V_total = 1.5 + 1.5 = 3.0 V

Three such cells provide:

4.5 V

A greater supply voltage can produce a greater current if the resistance remains unchanged.

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Analyzing an Unknown Series Circuit

Suppose:

V_supply = 18 V

R₁ = 3 Ω

R₂ = 6 Ω

R₃ = 9 Ω

Calculate equivalent resistance:

R_total = 3 + 6 + 9

R_total = 18 Ω

Calculate current:

I = 18/18

I = 1 A

Therefore:

I₁ = I₂ = I₃ = 1 A

Voltage drops:

V₁ = 1 × 3 = 3 V

V₂ = 1 × 6 = 6 V

V₃ = 1 × 9 = 9 V

Check:

3 + 6 + 9 = 18 V

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Finding an Unknown Resistance

A 20 V supply is connected to two series resistors.

The current is:

I = 2 A

One resistor has:

R₁ = 4 Ω

First calculate the total resistance:

R_total = V/I

R_total = 20/2

R_total = 10 Ω

Since:

R_total = R₁ + R₂

10 = 4 + R₂

Therefore:

R₂ = 6 Ω

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Finding an Unknown Voltage

Three resistors are connected in series.

The supply voltage is:

V_total = 18 V

Two measured voltage drops are:

V₁ = 5 V

V₂ = 7 V

Find V₃.

Use:

V_total = V₁ + V₂ + V₃

18 = 5 + 7 + V₃

18 = 12 + V₃

Therefore:

V₃ = 6 V

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Finding an Unknown Current

A 9 V battery is connected to two series resistors:

R₁ = 10 Ω

R₂ = 20 Ω

Total resistance:

R_total = 10 + 20

R_total = 30 Ω

Current:

I = V/R

I = 9/30

I = 0.30 A

Because it is a series circuit:

I₁ = I₂ = 0.30 A

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Power in Series Circuits

Electrical power can be calculated using:

P = VI

or:

P = I²R

Since the current is the same through series resistors, a resistor with greater resistance dissipates more power:

P = I²R

For the same current:

larger R → larger P

This also explains why the higher-resistance resistor has the larger voltage drop.

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Example: Comparing Power

Two series resistors carry a current of 2 A.

R₁ = 3 Ω

R₂ = 6 Ω

For R₁:

P₁ = I²R₁

P₁ = 2² × 3

P₁ = 12 W

For R₂:

P₂ = 2² × 6

P₂ = 24 W

The 6 Ω resistor transfers twice as much energy per second.

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Investigating Series Circuits Experimentally

A laboratory investigation can test the theoretical rules for series circuits.

Students might construct a circuit containing:

• power supply
• two resistors
• ammeter
• voltmeter
• switch

Measurements can then be made for:

• current before the first resistor
• current between the resistors
• current after the second resistor
• voltage across each resistor
• total supply voltage

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Expected Experimental Results

Theory predicts:

I₁ = I₂ = I₃

and:

V_supply = V₁ + V₂ + V₃

It also predicts:

R_total = R₁ + R₂ + R₃

Real experimental values should be close to these predictions.

However, they may not be exactly equal.

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Comparing Theory and Experiment

Suppose theory predicts:

I = 0.500 A

but the experiment gives:

I = 0.487 A

The values are close, but not identical.

Possible reasons include:

• resistor tolerances
• internal resistance of the power supply
• resistance of connecting wires
• contact resistance
• temperature changes
• meter uncertainty
• reading uncertainty

Real circuits are not perfectly ideal.

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Resistor Tolerance

A resistor labelled:

100 Ω ± 5%

does not necessarily have a resistance of exactly 100 Ω.

Its actual resistance could be approximately between:

95 Ω and 105 Ω.

Therefore, theoretical calculations using exactly 100 Ω may differ slightly from experimental measurements.

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Percentage Difference

A useful way to compare experimental and theoretical values is:

percentage difference = |experimental − theoretical| / theoretical × 100%

Suppose:

theoretical current = 0.500 A

experimental current = 0.487 A

Difference:

|0.487 − 0.500| = 0.013 A

Percentage difference:

0.013/0.500 × 100%

= 2.6%

A relatively small percentage difference suggests that the experimental result agrees reasonably well with the theoretical prediction.

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Example: Experimental Voltage Measurements

Suppose a 12 V supply is connected to three resistors.

Experimental voltage measurements are:

V₁ = 2.1 V

V₂ = 3.8 V

V₃ = 6.0 V

Add them:

V_total = 2.1 + 3.8 + 6.0

V_total = 11.9 V

The measured supply may be approximately 12.0 V.

The 0.1 V difference could result from experimental uncertainty.

The measurements therefore provide good evidence for:

V_supply ≈ V₁ + V₂ + V₃

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Ideal vs Real Circuits

In theoretical problems, we often assume:

• wires have zero resistance
• meters are perfect
• batteries maintain exactly their stated voltage
• resistors have exact resistance
• temperature remains constant

Real circuits do not perfectly satisfy these assumptions.

Therefore:

theoretical result ≈ experimental result

is often more realistic than expecting exact equality.

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Kirchhoff's Rules and Series Circuits

The basic rules of series circuits are consistent with more general circuit laws.

Conservation of charge explains why the current is the same throughout a series circuit.

Charge entering a component cannot simply disappear.

Conservation of energy explains why the voltage drops add to the supply voltage.

The energy supplied to each unit of charge is transferred among the components as the charge moves around the circuit.

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Series Circuit Problem-Solving Strategy

When analyzing a series circuit:

Identify the components

Determine which resistors are connected in series.

Find total resistance

Use:

R_total = R₁ + R₂ + R₃ + ...

Find circuit current

Use:

I = V_total/R_total

Use the same current everywhere

For a series circuit:

I₁ = I₂ = I₃

Find voltage drops

Use:

V = IR

for each resistor.

Check the voltage

Make sure:

V₁ + V₂ + V₃ = V_supply

This final check is very useful for detecting calculation errors.

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Complete Worked Example

A 30 V power supply is connected to three resistors in series:

R₁ = 3 Ω

R₂ = 5 Ω

R₃ = 7 Ω

Total resistance

R_total = 3 + 5 + 7

R_total = 15 Ω

Circuit current

I = V/R

I = 30/15

I = 2 A

Therefore:

I₁ = I₂ = I₃ = 2 A

Voltage across R₁

V₁ = 2 × 3

V₁ = 6 V

Voltage across R₂

V₂ = 2 × 5

V₂ = 10 V

Voltage across R₃

V₃ = 2 × 7

V₃ = 14 V

Check

V_total = 6 + 10 + 14

V_total = 30 V

The results satisfy all three major series circuit rules.

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Predicting Changes

Suppose the supply voltage remains constant.

If another resistor is added in series:

total resistance increases

Therefore:

current decreases

Since the current through every component decreases, the voltage drops across the original resistors may also change.

For identical lamps, this generally means:

adding lamps in series → lamps become dimmer

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Removing a Resistor

If a resistor is removed but the circuit is reconnected so that it remains closed:

total resistance decreases

For constant supply voltage:

current increases

If instead the component is simply removed and leaves a gap:

the circuit becomes open and current becomes zero.

These are two different situations and should not be confused.

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Common Mistakes

Mistake: Thinking current is divided between series components.
Current is the same throughout a series circuit.

Mistake: Thinking voltage is the same across every series component.
The supply voltage is shared according to resistance.

Mistake: Forgetting to add resistances.
For series resistors:

R_total = R₁ + R₂ + ...

Mistake: Saying current is used up.
Charge is conserved. Energy is transferred, but current does not gradually disappear around the circuit.

Mistake: Assuming voltage divides equally.
Voltage divides equally only when the resistances are equal.

Mistake: Adding another series resistor and predicting greater current.
Adding resistance decreases current if the supply voltage remains constant.

Mistake: Expecting experimental measurements to match theory perfectly.
Real components and measuring instruments have uncertainties and non-ideal behavior.

Mistake: Connecting a voltmeter in series.
Voltmeters are connected in parallel across components.

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Key Terms

Series circuit — circuit in which components form one continuous current path.

Current — rate of flow of electric charge.

Potential difference — energy transferred per unit charge.

Voltage drop — potential difference across a circuit component.

Resistance — opposition to electric current.

Equivalent resistance — single resistance that has the same overall effect as several resistors.

Ohm's Law — relationship V = IR for an ohmic conductor under constant physical conditions.

Ammeter — instrument used to measure current and connected in series.

Voltmeter — instrument used to measure potential difference and connected in parallel.

Voltage divider — series arrangement in which the supply voltage is divided among resistors.

Open circuit — incomplete circuit in which current cannot flow around the complete path.

Resistor tolerance — permitted difference between a resistor's labelled and actual resistance.

Experimental uncertainty — range of possible uncertainty associated with a measured value.

Theoretical value — value predicted using a mathematical model or physical law.

Experimental value — value obtained through measurement.

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Key Takeaways

• A series circuit contains only one path for electric current.
• Current is the same throughout a series circuit.
• I_total = I₁ = I₂ = I₃
• The supply voltage is shared between series components.
• V_total = V₁ + V₂ + V₃
• Series resistances add together.
• R_total = R₁ + R₂ + R₃
• Equivalent resistance represents the combined effect of all the resistors.
• Adding another resistor in series increases total resistance.
• If supply voltage remains constant, increasing resistance decreases current.
• Larger series resistors have larger voltage drops because the same current passes through them.
• Equal resistors divide the supply voltage equally.
• Ohm's Law can be combined with series circuit rules to calculate unknown values.
• A break anywhere in a series circuit stops current everywhere.
• Voltage division can be used to predict the voltage across individual resistors.
• Experimental results should be compared with theoretical predictions.
• Small differences between theory and experiment can result from component tolerances, temperature changes and measurement uncertainty.
• The constant current in a series circuit is consistent with conservation of charge.
• The sum of voltage drops equalling the supply voltage is consistent with conservation of energy.

Check Your Understanding

1. What is meant by a series circuit?
2. Describe the current at different points in a series circuit.
3. Three resistors carry a current of 0.8 A in series. What is the current through each resistor?
4. State the equation for total resistance in a series circuit.
5. Calculate the equivalent resistance of 3 Ω, 7 Ω and 10 Ω connected in series.
6. Four identical 15 Ω resistors are connected in series. Calculate the total resistance.
7. A 12 V supply is connected to a total resistance of 6 Ω. Calculate the current.
8. Two resistors of 2 Ω and 4 Ω are connected in series across 12 V. Calculate the circuit current.
9. For Question 8, calculate the voltage across each resistor.
10. Explain why the 4 Ω resistor in Question 8 has a larger voltage drop.
11. Three identical resistors are connected in series across 18 V. What is the voltage across each resistor?
12. A 5 Ω resistor and a 10 Ω resistor are connected in series across 30 V. Calculate the total resistance, current and voltage across each resistor.
13. Predict what happens to total resistance when another resistor is added in series.
14. Predict what happens to current when another resistor is added while supply voltage remains constant.
15. Explain why identical lamps generally become dimmer when more are connected in series.
16. Explain what happens if one lamp in a simple series circuit breaks.
17. A theoretical current is 0.80 A and an experimental current is 0.76 A. Calculate the percentage difference.
18. Give three reasons why experimental and theoretical circuit results may differ slightly.
19. Explain why the current rule for series circuits is connected to conservation of charge.
20. Explain why the sum of the voltage drops should equal the supply voltage.