Parallel Circuit: Definition, Rules, Formulas, and Examples

A parallel circuit is an electrical circuit in which components are connected across the same two points, creating more than one path for electric current. Because each path is connected to the same two points, every branch has the same voltage across it. The current divides among the branches and then combines again. ([openstax.org](https://openstax.org/books/physics/pages/19-3-parallel-circuits?utm_source=openai))

What Is a Parallel Circuit?

In a parallel circuit, each component is placed on its own branch. The branches are connected between the same two nodes, or connection points.

For example, several lamps connected across the same battery terminals form a parallel circuit. Each lamp has its own path for current.

Main Rules of a Parallel Circuit

1. Voltage is the same across every branch

The voltage across each parallel component is equal to the voltage supplied by the source:

V_total = V₁ = V₂ = V₃

For example, if a 12 V battery is connected to three resistors in parallel, each resistor has 12 V across it.

2. Total current is the sum of branch currents

Current divides when it reaches a junction. The total current supplied by the source equals the sum of the currents in all branches:

I_total = I₁ + I₂ + I₃ + …

This rule follows the conservation of electric charge. Current is not created or lost at a junction. ([openstax.org](https://openstax.org/books/physics/pages/19-3-parallel-circuits?utm_source=openai))

3. Equivalent resistance is lower

The total or equivalent resistance of parallel resistors is less than the smallest individual resistance in the group:

R_eq < the smallest branch resistance

Adding another branch gives current an additional path, so the circuit offers less overall opposition to current. ([openstax.org](https://openstax.org/books/university-physics-volume-2/pages/10-2-resistors-in-series-and-parallel?utm_source=openai))

Parallel Circuit Formulas

Ohm’s law

Ohm’s law connects voltage, current, and resistance:

V = IR

The formula can also be rearranged as:

• I = V ÷ R
• R = V ÷ I

In a parallel circuit, use the voltage across a particular branch and the resistance of that branch to find its current. ([openstax.org](https://openstax.org/books/college-physics-2e/pages/20-2-ohms-law-resistance-and-simple-circuits?utm_source=openai))

Equivalent resistance of two resistors

For two resistors connected in parallel:

R_eq = (R₁ × R₂) ÷ (R₁ + R₂)

Equivalent resistance of three or more resistors

For any number of parallel resistors:

1 ÷ R_eq = 1 ÷ R₁ + 1 ÷ R₂ + 1 ÷ R₃ + …

Therefore:

R_eq = 1 ÷ (1 ÷ R₁ + 1 ÷ R₂ + 1 ÷ R₃ + …)

This reciprocal formula is the general rule for resistors in parallel. ([openstax.org](https://openstax.org/books/physics/pages/19-3-parallel-circuits?utm_source=openai))

How Current Divides in Parallel Branches

Since every branch has the same voltage, the branch current depends on the branch resistance:

I_branch = V ÷ R_branch

A branch with lower resistance carries more current. A branch with higher resistance carries less current.

If two parallel branches have equal resistance, they carry equal currents.

Worked Numerical Example

Suppose a 12 V source is connected to two resistors in parallel:

• R₁ = 6 Ω
• R₂ = 3 Ω

Step 1: Find the equivalent resistance

Use the formula for two parallel resistors:

R_eq = (6 × 3) ÷ (6 + 3)

R_eq = 18 ÷ 9 = 2 Ω

Step 2: Find the total current

Use Ohm’s law:

I_total = V ÷ R_eq

I_total = 12 V ÷ 2 Ω = 6 A

Step 3: Find the current in each branch

Each resistor has the full 12 V across it.

I₁ = 12 V ÷ 6 Ω = 2 A

I₂ = 12 V ÷ 3 Ω = 4 A

Step 4: Check the current rule

I_total = I₁ + I₂

6 A = 2 A + 4 A

The result is correct. The 3 Ω resistor carries more current because it has the lower resistance.

Parallel Circuit Compared with a Series Circuit

Advantages of Parallel Circuits

• Each branch receives the full source voltage.
• One branch can continue working if another branch is opened, provided the source and remaining branches are still connected.
• Components can operate independently.
• Additional branches can be added without changing the voltage across the existing branches in an ideal source circuit.

Common Applications

Parallel connections are useful when several devices need to operate independently from the same voltage source. Examples include many lighting and electrical distribution systems, electronic circuits, and groups of loads connected across a supply.

Real circuits also contain wires, switches, and sources with limits. Therefore, practical circuits must be designed so that the source, conductors, and protective devices can safely handle the total current.

Common Mistakes and Confusions

1. Assuming current is the same in every branch: In parallel circuits, voltage is the same across branches, but branch currents can be different.
2. Adding parallel resistances directly: Direct addition is used for series resistors. Parallel resistors require the reciprocal formula.
3. Forgetting that total current is larger than a branch current: The source current equals the sum of all branch currents.
4. Using the wrong voltage in Ohm’s law: For a branch current, use the voltage across that branch and the resistance of that branch.
5. Not checking the answer: The equivalent resistance of parallel resistors must be smaller than the smallest individual resistance.

Important Points to Remember

• A parallel circuit has multiple current paths.
• All parallel branches have the same voltage.
• Total current equals the sum of branch currents.
• A lower-resistance branch carries more current.
• The equivalent resistance is less than the smallest individual resistance.
• Use Ohm’s law, V = IR, to calculate unknown voltage, current, or resistance.