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Resistors in Parallel

Total resistance of components side⁻by-side.

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

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Core idea

Overview

In a parallel circuit, multiple resistors are connected across the same two nodes, ensuring that each component experiences the same potential difference or voltage. The reciprocal of the total equivalent resistance is the sum of the reciprocals of the individual resistances, which effectively increases the total path area for current and decreases overall resistance.

When to use: Apply this equation when electrical components are configured in separate branches so that the current splits between them. It assumes an ideal circuit where the connecting wires have zero resistance and the voltage remains constant across all parallel branches.

Why it matters: This principle is the foundation of modern electrical distribution, such as household wiring, where it allows devices to operate independently at a standard voltage. It also enables engineers to combine standard resistor values to achieve specific, non-standard resistance levels required for sensitive electronics.

Symbols

Variables

R_T = Total Resistance, R_1 = Resistor 1, R_2 = Resistor 2

R_{T}

Total Resistance

Ω

R_{1}

Resistor 1

Ω

R_{2}

Resistor 2

Ω

Walkthrough

Derivation

Derivation of Resistors in Parallel

Proves that the reciprocal of total resistance in a parallel circuit is the sum of the reciprocals of individual resistances.

The connecting wires have zero resistance.
The components obey Ohm's Law.

Apply Kirchhoff's Current Law:

In a parallel circuit, the total current splits into the parallel branches.

I_{total} = I_{1} + I_{2} + I_{3}

Apply Ohm's Law:

Substitute I = V/R. Because it's a parallel circuit, the potential difference V is the same across all branches.

\frac{V}{R _{total}} = \frac{V}{R _{1}} + \frac{V}{R _{2}} + \frac{V}{R _{3}}

Cancel the Voltage:

Divide the entire equation by the common voltage V.

\frac{1}{R _{total}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \frac{1}{R _{3}}

Note: For two resistors: $R_{total} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$ .

Result

\frac{1}{R _{total}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \frac{1}{R _{3}}

Source: AQA A-Level Physics — Current Electricity

Free formulas

Rearrangements

Solve for $R_{T}$

Make RT the subject

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

Rearrange the formula for resistors in parallel to solve for the total resistance, $R_{T}$ .

Difficulty: 3/5

Solve for $R_{1}$

Make R1 the subject

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

Start with the formula for resistors in parallel. Isolate the term with R1, combine fractions, then take the reciprocal to solve for R1.

Difficulty: 2/5

Solve for $R_{2}$

Make R2 the subject

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

Rearrange the formula for resistors in parallel to solve for R2.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph forms a hyperbola because R1 appears in the denominator of the rearranged formula. As R1 increases, Rt approaches R2 as a horizontal asymptote, with the curve restricted to values where R1 is greater than Rt. In physical terms, this means that adding a very large resistor in parallel has little effect on the total resistance, while a small resistor significantly pulls the total resistance down toward its own value. The most important feature is that the curve never reaches zero, meaning that adding any fi

Graph type: hyperbolic

Why it behaves this way

Intuition

Imagine a multi-lane highway where each lane (representing an individual resistor) has a different capacity for traffic flow; the total traffic capacity of the highway increases with each additional lane, making it

R_{T}

The total equivalent resistance of the entire parallel combination.

It represents the overall opposition to electrical current flow when multiple paths are available. Since adding more parallel paths makes it easier for current to flow,

R_{T}

will always be less than any individual

R_{i}

R_{1}

The individual electrical resistance of the first component in the parallel circuit.

Each

R_{i}

quantifies how much that specific path opposes current flow. A higher

R_{i}

means that particular path is more restrictive.

R_{2}

The individual electrical resistance of the second component in the parallel circuit.

Each

R_{i}

quantifies how much that specific path opposes current flow. A higher

R_{i}

means that particular path is more restrictive.

Signs and relationships

1/R: The reciprocal of resistance, often called conductance (G), represents how easily current flows through a component. In parallel circuits, conductances add up because each path provides an additional way for current to

Free study cues

Insight

Canonical usage

Resistance values are typically expressed in Ohms (Ω) within the International System of Units (SI).

Common confusion

A common mistake is forgetting to take the reciprocal of the total equivalent resistance (1/ $R_{T}$ ) after summing the reciprocals of the individual resistances, or mixing units (e.g., Ohms and kilo-Ohms)

Unit systems

$R$ Ohm (Ω) · All resistance values (R_T, R_1, R_2, etc.) must be expressed in the same unit for the equation to be dimensionally consistent. The unit of 1/R is Siemens (S) or reciprocal Ohms (Ω-1).

One free problem

Practice Problem

A 10 Ω resistor and a second 10 Ω resistor are connected in parallel. Calculate the total equivalent resistance of this circuit branch.

Resistor 110 Ω

Resistor 210 Ω

Solve for: $R t$

Hint: When resistors are identical, the total resistance is simply the resistance value divided by the number of resistors.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the equivalent resistance of two parallel bulbs.

Study smarter

Tips

The total equivalent resistance (Rt) will always be smaller than the smallest individual resistor in the group.
Remember to take the reciprocal of your final sum to solve for Rt, rather than leaving it as 1/Rt.
If you have two resistors of identical value, the total resistance is exactly half of that value.
Ensure all resistance units are consistent (e.g., all in Ohms or all in kOhms) before starting the calculation.

Avoid these traps

Common Mistakes

Using the series sum.
Mixing up R1 and R2 in rearrangements.

Common questions

Frequently Asked Questions

Proves that the reciprocal of total resistance in a parallel circuit is the sum of the reciprocals of individual resistances.

Apply this equation when electrical components are configured in separate branches so that the current splits between them. It assumes an ideal circuit where the connecting wires have zero resistance and the voltage remains constant across all parallel branches.

This principle is the foundation of modern electrical distribution, such as household wiring, where it allows devices to operate independently at a standard voltage. It also enables engineers to combine standard resistor values to achieve specific, non-standard resistance levels required for sensitive electronics.

Using the series sum. Mixing up R1 and R2 in rearrangements.

Calculating the equivalent resistance of two parallel bulbs.

The total equivalent resistance (Rt) will always be smaller than the smallest individual resistor in the group. Remember to take the reciprocal of your final sum to solve for Rt, rather than leaving it as 1/Rt. If you have two resistors of identical value, the total resistance is exactly half of that value. Ensure all resistance units are consistent (e.g., all in Ohms or all in kOhms) before starting the calculation.

References

Sources

Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). John Wiley & Sons.
Wikipedia: Series and parallel circuits
Halliday, Resnick, Walker, Fundamentals of Physics, 10th ed.
Wikipedia: Ohm (unit)
Halliday, Resnick, Walker Fundamentals of Physics
Young and Freedman University Physics
AQA A-Level Physics — Current Electricity

Resistors in Parallel

Overview

Variables

Derivation

Apply Kirchhoff's Current Law:

Apply Ohm's Law:

Cancel the Voltage:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Resistors in Series

Frequently Asked Questions

Sources