PhysicsElectricity and Circuits

Resistance (Parallel)

In a parallel circuit, the total resistance is always less than the smallest individual resistor. Adding more parallel paths provides more room for current to flow, effectively decreasing the overall opposition to the current. This relationship is defined by the reciprocal of the total resistance being equal to the sum of the reciprocals of each individual resistor.

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

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Core idea

Overview

When to use: Use this when you need to find the total resistance of two or more resistors connected in parallel branches across a voltage source.

Why it matters: It explains why adding more appliances to a domestic parallel circuit reduces the total resistance, allowing more current to be drawn from the power supply.

Remember it

Memory Aid

Phrase: One over the Total is the sum of Ones over individual parts.

Visual Analogy: Think of a busy supermarket: one checkout lane (Total resistance) handles the flow much faster when you open multiple lanes (R1, R2), so the 'reciprocal' of the total flow is just the sum of the flow in each individual lane.

Exam Tip: Don't forget to flip your final answer! After calculating the sum of the fractions, you have found 1/RT, so you must press the 'x⁻¹' button on your calculator to find the actual value of RT.

Why it makes sense

Intuition

Imagine a multi-lane highway where each resistor is a toll booth lane. The 'conductivity' (the ease of flow) is the width of a lane. Adding more lanes in parallel adds more paths for traffic, increasing the total 'width' (conductivity) of the road, even if each individual lane is narrow.

Symbols

Variables

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

R_{T}

Total Resistance

V a r iab l e

R_{1}

Resistor 1

V a r iab l e

R_{2}

Resistor 2

V a r iab l e

Walkthrough

Derivation

Derivation of Resistance (Parallel)

This derivation utilizes Kirchhoff's Current Law and Ohm's Law to relate the total resistance of parallel branches to the individual resistances.

The power source provides a constant voltage V across all parallel branches.
The connecting wires have negligible resistance.

Kirchhoff's Current Law

In a parallel circuit, the total current entering a junction is equal to the sum of the currents passing through each branch.

I_{T} = I_{1} + I_{2} + \dots

Note: Remember that current 'splits' at junctions in parallel circuits.

Apply Ohm's Law

We use Ohm's Law (I = V/R) to express each branch current in terms of the total voltage V and the branch resistance.

I = \frac{V}{R}

Note: Since components are in parallel, the voltage V is the same for every resistor.

Substitute and Simplify

Substitute the Ohm's Law expressions into the Kirchhoff's Current Law equation.

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

Note: The voltage V is common to all terms.

Final Form

Divide both sides of the equation by V to isolate the reciprocal of total resistance.

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

Note: Always remember to invert your final answer (1/ $R_{T}$ ) to find $R_{T}$ .

Result

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

Source: AQA GCSE Physics Specification

Where it shows up

Real-World Context

In household wiring, lights and appliances are connected in parallel so that if one bulb breaks, the others remain powered independently.

Avoid these traps

Common Mistakes

Forgetting to find the reciprocal of the final sum; students often leave the answer as 1/ $R_{T}$ .
Assuming the total resistance in parallel is simply the sum of resistors (which is true only for series circuits).

Study smarter

Tips

Always calculate the sum of the reciprocals before taking the final reciprocal to find $R_{t}$ otal.
Remember that the unit of resistance is Ohms (Ω).
If you have only two resistors, you can use the shortcut: $R_{T}$ = (R1 * R2) / (R1 + R2).

Common questions

Frequently Asked Questions

This derivation utilizes Kirchhoff's Current Law and Ohm's Law to relate the total resistance of parallel branches to the individual resistances.

Use this when you need to find the total resistance of two or more resistors connected in parallel branches across a voltage source.

It explains why adding more appliances to a domestic parallel circuit reduces the total resistance, allowing more current to be drawn from the power supply.

Forgetting to find the reciprocal of the final sum; students often leave the answer as 1/R_T. Assuming the total resistance in parallel is simply the sum of resistors (which is true only for series circuits).

In household wiring, lights and appliances are connected in parallel so that if one bulb breaks, the others remain powered independently.

Always calculate the sum of the reciprocals before taking the final reciprocal to find R_total. Remember that the unit of resistance is Ohms (Ω). If you have only two resistors, you can use the shortcut: R_T = (R1 * R2) / (R1 + R2).