4. Parallel Circuits

Parallel Circuits

What Is a Parallel Circuit?

A parallel circuit contains two or more separate paths, called branches, through which electric current can flow.

At a junction, current can divide between the branches. The currents then recombine when the branches join again.

The main rules for a parallel circuit are:

Voltage is the same across each parallel branch.

Current divides between the branches.

The total current equals the sum of the branch currents.

Adding parallel branches decreases the equivalent resistance.

These rules make parallel circuits behave very differently from series circuits.

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Voltage in a Parallel Circuit

Components connected in parallel are connected between the same two points in the circuit.

Therefore, they have the same potential difference.

For parallel branches:

V_total = V₁ = V₂ = V₃ = ...

Suppose two resistors are connected in parallel across a 12 V battery.

Then:

V₁ = 12 V

V₂ = 12 V

The voltage is not divided between the branches.

Each branch receives the full supply voltage.

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Why Is the Voltage the Same?

Potential difference measures the energy transferred per unit charge between two points.

Parallel branches begin and end at the same two electrical points.

Therefore, the potential difference between the ends of each branch must be the same.

If three lamps are connected in parallel across a 9 V supply, each lamp has approximately:

9 V across it

This is one major advantage of parallel circuits.

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Measuring Voltage

A voltmeter is connected in parallel across the component whose potential difference is being measured.

If two resistors are connected in parallel across a 6 V supply, measurements should show approximately:

V_supply = 6 V

V₁ = 6 V

V₂ = 6 V

Small differences may occur in a real experiment because of measurement uncertainty and non-ideal components.

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Current in a Parallel Circuit

Current behaves differently from voltage.

At a junction, the current splits between the available branches.

The total current entering a junction must equal the total current leaving it.

For a simple parallel circuit:

I_total = I₁ + I₂ + I₃ + ...

For example:

I₁ = 2 A

I₂ = 3 A

Then:

I_total = 2 + 3

I_total = 5 A

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Conservation of Charge

The current rule for parallel circuits follows from the conservation of charge.

Charge cannot simply disappear at a junction.

If 5 coulombs of charge arrive at a junction each second, a total of 5 coulombs per second must leave through the branches.

For example:

3 C/s may travel through one branch

and:

2 C/s may travel through another.

Since:

1 A = 1 C/s

the currents are:

3 A + 2 A = 5 A

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Current Division

Current does not necessarily divide equally between parallel branches.

The amount of current in each branch depends on the branch resistance.

Using Ohm's Law:

I = V/R

Since each branch has the same voltage:

lower resistance → greater current

higher resistance → smaller current

This is called current division.

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Example of Current Division

Two resistors are connected in parallel across 12 V.

R₁ = 4 Ω

R₂ = 6 Ω

Current through R₁:

I₁ = V/R₁

I₁ = 12/4

I₁ = 3 A

Current through R₂:

I₂ = 12/6

I₂ = 2 A

Total current:

I_total = I₁ + I₂

I_total = 3 + 2

I_total = 5 A

Notice that the lower-resistance branch carries the larger current.

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Equal Resistances

If two identical resistors are connected in parallel, the current divides equally.

Suppose two 10 Ω resistors are connected across 20 V.

Current through each resistor:

I = V/R

I = 20/10

I = 2 A

Therefore:

I₁ = 2 A

I₂ = 2 A

Total current:

I_total = 4 A

Equal resistances produce equal branch currents.

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Measuring Current

Current is measured using an ammeter.

An ammeter must be connected in series with the path whose current is being measured.

An ammeter placed before the junction measures:

total current

An ammeter placed inside one branch measures:

branch current

An ammeter after the branches recombine again measures:

total current

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Equivalent Resistance in Parallel

Several parallel resistors can be replaced mathematically by a single equivalent resistance.

For resistors in parallel:

1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...

This rule is very different from the series resistance rule.

In parallel:

the equivalent resistance is always less than the smallest individual branch resistance.

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Calculating Equivalent Resistance

Suppose:

R₁ = 6 Ω

R₂ = 3 Ω

Then:

1/R_total = 1/6 + 1/3

Convert to a common denominator:

1/R_total = 1/6 + 2/6

1/R_total = 3/6

1/R_total = 1/2

Therefore:

R_total = 2 Ω

Notice:

2 Ω is smaller than both 3 Ω and 6 Ω.

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Why Does Parallel Resistance Decrease?

Adding a parallel branch gives charge another path through the circuit.

More pathways allow more total current to flow for the same supply voltage.

From Ohm's Law:

R = V/I

If V remains constant while total I increases, the equivalent resistance must decrease.

Therefore:

more parallel paths → lower equivalent resistance

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Two-Resistor Shortcut

For exactly two resistors in parallel:

R_total = (R₁R₂)/(R₁ + R₂)

For example:

R₁ = 4 Ω

R₂ = 12 Ω

Then:

R_total = (4 × 12)/(4 + 12)

R_total = 48/16

R_total = 3 Ω

This shortcut works only for two parallel resistors.

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Equal Resistors in Parallel

If n identical resistors, each with resistance R, are connected in parallel:

R_total = R/n

For example, three identical 12 Ω resistors are connected in parallel.

R_total = 12/3

R_total = 4 Ω

Four identical 20 Ω resistors would give:

R_total = 20/4

R_total = 5 Ω

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Checking Your Parallel Resistance Answer

A useful check is:

R_total must be smaller than the smallest branch resistance.

Suppose you calculate the equivalent resistance of:

10 Ω and 15 Ω

and obtain:

25 Ω

That cannot be correct.

Since the resistors are in parallel, the answer must be:

less than 10 Ω

The correct calculation is:

R_total = (10 × 15)/(10 + 15)

R_total = 150/25

R_total = 6 Ω

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Analyzing a Complete Parallel Circuit

Suppose two resistors are connected in parallel across 12 V:

R₁ = 6 Ω

R₂ = 4 Ω

Voltage

Because the resistors are parallel:

V₁ = V₂ = 12 V

Branch currents

I₁ = 12/6

I₁ = 2 A

I₂ = 12/4

I₂ = 3 A

Total current

I_total = 2 + 3

I_total = 5 A

Equivalent resistance

R_total = V_total/I_total

R_total = 12/5

R_total = 2.4 Ω

Check using the parallel formula:

1/R_total = 1/6 + 1/4

1/R_total = 2/12 + 3/12

1/R_total = 5/12

R_total = 12/5

R_total = 2.4 Ω

Both methods agree.

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Three Resistors in Parallel

Consider:

V = 12 V

R₁ = 4 Ω

R₂ = 6 Ω

R₃ = 12 Ω

Since all branches have the same voltage:

V₁ = V₂ = V₃ = 12 V

Calculate branch currents:

I₁ = 12/4 = 3 A

I₂ = 12/6 = 2 A

I₃ = 12/12 = 1 A

Total current:

I_total = 3 + 2 + 1

I_total = 6 A

Equivalent resistance:

R_total = V/I

R_total = 12/6

R_total = 2 Ω

Again, the equivalent resistance is smaller than the smallest branch resistance of 4 Ω.

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Predicting the Effect of Adding a Branch

Suppose a battery maintains a constant voltage.

Adding another resistor in parallel:

• creates another current path
• decreases equivalent resistance
• increases total current supplied by the battery

This is the opposite of adding another resistor in series.

Series

Add resistor → R_total increases → I_total decreases

Parallel

Add branch → R_total decreases → I_total increases

This is one of the most important comparisons between the two circuit types.

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What Happens to Existing Branches?

Suppose an ideal 12 V source has one 6 Ω resistor connected across it.

Current:

I = 12/6

I = 2 A

Now add another 6 Ω resistor in parallel.

Each branch still has:

12 V

Each branch therefore still carries:

2 A

But the total current becomes:

I_total = 2 + 2

I_total = 4 A

So adding the second branch increases the current supplied by the source without changing the current through the original resistor, assuming an ideal constant-voltage source.

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Lamps in Parallel

When identical lamps are connected in parallel:

• each lamp receives the full supply voltage
• each lamp can operate approximately as it would alone
• adding another lamp increases the total current
• removing one lamp does not necessarily turn off the others

This is very different from a series circuit.

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If One Parallel Branch Breaks

Suppose three lamps are connected in parallel.

If one lamp fails, that branch becomes open.

Current stops through that branch.

However, the other branches still provide complete paths through the circuit.

Therefore, the other lamps can continue operating.

This is an important advantage of parallel circuits.

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Household Circuits

Electrical devices in buildings are connected mainly in parallel.

This means different appliances receive approximately the same supply voltage and can operate independently.

Switching off one device does not normally interrupt the current through all the others.

Parallel connections therefore provide much greater independence than a single series circuit.

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Comparing Series and Parallel Circuits

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Current Division and Resistance

For two parallel resistors:

R₁ and R₂

the currents are:

I₁ = V/R₁

I₂ = V/R₂

Since V is the same:

I₁/I₂ = R₂/R₁

This means the current ratio is the inverse of the resistance ratio.

For example:

R₁ = 3 Ω

R₂ = 6 Ω

Then:

I₁/I₂ = 6/3

I₁/I₂ = 2

So the 3 Ω branch carries twice the current of the 6 Ω branch.

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Worked Current Division Example

A total current of 6 A enters two parallel branches.

The resistances are:

R₁ = 4 Ω

R₂ = 8 Ω

The 4 Ω branch has half the resistance, so it carries twice the current.

The currents must add to 6 A.

Therefore:

I₁ = 4 A

I₂ = 2 A

Check:

4 + 2 = 6 A

We can also determine the branch voltage:

V = I₁R₁

V = 4 × 4

V = 16 V

Check the second branch:

V = 2 × 8

V = 16 V

The voltage is the same across both branches, as expected.

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Finding an Unknown Branch Current

A total current of 8 A reaches a junction.

One branch carries 3 A.

Another carries 2 A.

A third branch carries I₃.

Use:

I_total = I₁ + I₂ + I₃

8 = 3 + 2 + I₃

8 = 5 + I₃

Therefore:

I₃ = 3 A

This is an application of conservation of charge.

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Finding an Unknown Resistance

Two resistors are connected in parallel across 12 V.

The total current is 5 A.

One branch contains:

R₁ = 4 Ω

First calculate current through R₁:

I₁ = 12/4

I₁ = 3 A

Total current:

I_total = I₁ + I₂

5 = 3 + I₂

Therefore:

I₂ = 2 A

Now:

R₂ = V/I₂

R₂ = 12/2

R₂ = 6 Ω

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Mixed Series-Parallel Networks

Some circuits contain both series and parallel connections.

The best strategy is to simplify the circuit one section at a time.

For example, suppose:

R₁ = 4 Ω

R₂ = 6 Ω

are connected in parallel, and this combination is connected in series with:

R₃ = 5 Ω

First find the parallel equivalent:

R_parallel = (4 × 6)/(4 + 6)

R_parallel = 24/10

R_parallel = 2.4 Ω

Now add the series resistor:

R_total = 2.4 + 5

R_total = 7.4 Ω

The entire network therefore behaves like a 7.4 Ω resistance.

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Strategy for Analyzing Parallel Networks

When solving a parallel circuit:

Identify the branches

Determine which components share the same two connection points.

Use equal voltage

For parallel branches:

V₁ = V₂ = V₃

Calculate branch currents

Use:

I = V/R

for each branch.

Add the currents

Use:

I_total = I₁ + I₂ + I₃

Calculate equivalent resistance

Use either:

1/R_total = 1/R₁ + 1/R₂ + ...

or:

R_total = V_total/I_total

Check your answer

Make sure:

R_total < smallest branch resistance

This is a very useful error check.

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Series-Parallel Analysis Strategy

For a mixed network:

1. Identify simple series or parallel groups.
2. Replace one group with its equivalent resistance.
3. Redraw or mentally simplify the circuit.
4. Continue until one total resistance remains.
5. Calculate total current.
6. Work backward through the circuit.
7. Use series and parallel rules to find individual currents and voltages.

This approach prevents complicated networks from becoming overwhelming.

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Power in Parallel Circuits

Electrical power can be calculated using:

P = VI

For a resistor:

P = V²/R

Since parallel branches have the same voltage, a lower-resistance branch dissipates more power.

For example, two resistors are connected across 12 V:

R₁ = 6 Ω

R₂ = 12 Ω

For R₁:

P₁ = 12²/6

P₁ = 144/6

P₁ = 24 W

For R₂:

P₂ = 144/12

P₂ = 12 W

The lower-resistance branch carries more current and transfers more energy per second.

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Experimental Investigation

A parallel circuit can be investigated using ammeters and voltmeters.

Measurements can test whether:

V_supply ≈ V₁ ≈ V₂

and:

I_total ≈ I₁ + I₂

Small differences may occur because of:

• meter uncertainty
• resistor tolerances
• connecting-wire resistance
• contact resistance
• internal resistance of the source
• temperature changes

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Parallel Circuits and Conservation Laws

The two major parallel circuit rules are connected to fundamental physical principles.

Conservation of charge

At a junction:

current entering = current leaving

This explains:

I_total = I₁ + I₂ + ...

Conservation of energy

Branches connected between the same two points have the same energy change per unit charge.

This explains:

V_total = V₁ = V₂ = ...

These rules are not arbitrary—they follow from fundamental conservation principles.

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Why Parallel Resistance Can Seem Strange

Students sometimes find it surprising that connecting two resistors produces an equivalent resistance smaller than either resistor.

Consider two 10 Ω resistors.

One resistor alone:

R = 10 Ω

Two in series:

R = 20 Ω

Two in parallel:

R = 5 Ω

The parallel arrangement has two simultaneous pathways for current.

This allows twice as much total current for the same voltage.

Since:

R = V/I

doubling current while maintaining voltage halves the equivalent resistance.

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Series vs Parallel Example

Suppose two identical 6 Ω resistors are connected to a 12 V source.

In series

R_total = 6 + 6

R_total = 12 Ω

I_total = 12/12

I_total = 1 A

Each resistor carries:

1 A

Each has:

6 V

across it.

In parallel

R_total = 6/2

R_total = 3 Ω

I_total = 12/3

I_total = 4 A

Each branch has:

12 V

Current through each branch:

I = 12/6

I = 2 A

The same components behave very differently depending on how they are connected.

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Common Mistakes

Mistake: Saying current is the same in every parallel branch.
Current divides according to branch resistance.

Mistake: Saying voltage divides between parallel branches.
Parallel branches have the same voltage.

Mistake: Adding parallel resistances directly.
The direct addition rule applies to series circuits.

Mistake: Obtaining an equivalent resistance larger than the smallest branch resistance.
Parallel equivalent resistance must be smaller than the smallest individual resistance.

Mistake: Assuming current divides equally.
It divides equally only when branch resistances are equal.

Mistake: Thinking current is lost at a junction.
Total current entering equals total current leaving.

Mistake: Thinking adding a parallel resistor increases total resistance.
Adding another parallel path decreases equivalent resistance.

Mistake: Applying the two-resistor shortcut to three or more resistors.
Use the reciprocal formula for larger networks.

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Key Terms

Parallel circuit — circuit containing two or more separate paths for current.

Branch — individual path within a parallel circuit.

Junction — point where current paths divide or recombine.

Branch current — current flowing through one parallel path.

Total current — combined current supplied to all branches.

Equivalent resistance — single resistance that produces the same overall circuit behavior as a network of resistors.

Current division — splitting of current among parallel branches.

Potential difference — energy transferred per unit charge.

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

Conservation of charge — principle that electric charge cannot be created or destroyed in an ordinary circuit.

Mixed circuit — circuit containing both series and parallel connections.

Open branch — branch with a break that prevents current from flowing through it.

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Key Takeaways

• Parallel circuits contain two or more current paths.
• Voltage is the same across parallel branches.
• V_total = V₁ = V₂ = V₃
• Current divides at junctions.
• I_total = I₁ + I₂ + I₃
• Lower-resistance branches carry greater current.
• Equal resistances carry equal currents when connected in parallel.
• Parallel equivalent resistance is calculated using the reciprocal rule.
• For two resistors, R_total = R₁R₂/(R₁ + R₂).
• The equivalent resistance is always smaller than the smallest parallel resistance.
• Adding another parallel branch decreases equivalent resistance.
• At constant supply voltage, adding a parallel branch increases total current.
• Existing ideal parallel branches continue to receive the full supply voltage.
• If one parallel branch breaks, the other branches can continue operating.
• Household electrical systems use parallel connections so devices can operate independently.
• Current division follows from Ohm's Law and the equal voltage across branches.
• The current rule follows from conservation of charge.
• Parallel networks can be combined with series sections to create mixed circuits.
• Complex networks can be analyzed by simplifying one section at a time.
• Series and parallel circuits behave very differently even when they contain the same components.

Check Your Understanding

1. What is meant by a parallel circuit?
2. Describe the voltage across different branches of a parallel circuit.
3. Describe what happens to current at a junction.
4. A total current of 8 A divides into branches carrying 3 A and 5 A. Show that charge is conserved.
5. A total current of 10 A divides into three branches. Two carry 2 A and 3 A. Calculate the third current.
6. Two resistors of 6 Ω and 3 Ω are connected in parallel. Calculate the equivalent resistance.
7. Two resistors of 4 Ω and 12 Ω are connected in parallel. Calculate the equivalent resistance.
8. Three identical 15 Ω resistors are connected in parallel. Calculate the equivalent resistance.
9. Why must parallel equivalent resistance be smaller than the smallest branch resistance?
10. Two resistors of 4 Ω and 8 Ω are connected across 16 V. Calculate the current through each branch.
11. For Question 10, calculate the total current.
12. Explain why the 4 Ω resistor carries more current than the 8 Ω resistor.
13. What happens to equivalent resistance when another parallel branch is added?
14. What happens to total current when another branch is added while supply voltage remains constant?
15. Explain what happens to the other lamps if one lamp in a parallel lighting circuit fails.
16. Compare voltage in series and parallel circuits.
17. Compare current in series and parallel circuits.
18. Compare the effect of adding a resistor in series with adding a resistor in parallel.
19. A 5 Ω resistor is connected in series with two 12 Ω resistors connected in parallel. Calculate the total equivalent resistance.
20. Explain why parallel circuits are generally more suitable than simple series circuits for connecting independent electrical devices.