# Resistors in Parallel

## The total resistance in a parallel circuit is equal to the sum of the inverse of each individual resistances.

#### Key Points

• The total resistance in a parallel circuit is less than the smallest of the individual resistances.

• Each resistor in parallel has the same voltage of the source applied to it (voltage is constant in a parallel circuit).

• Parallel resistors do not each get the total current; they divide it (current is dependent on the value of each resistor and the number of total resistors in a circuit).

#### Terms

• The opposition to the passage of an electric current through that element.

• An arrangement of electrical components such that a current flows along two or more paths.

#### Figures

1. ##### Resistors in Parallel

A parallel connection of resistors.

2. ##### Parallel resistors

Three resistors connected in parallel to a battery and the equivalent single or parallel resistance.

3. ##### Parallel Circuits

A brief overview of parallel circuit analysis using VIRP tables for high school physics students.

### Overview

Resistors in a circuit can be connected in series or in parallel. The total resistance of a combination of resistors depends on both their individual values and how they are connected.

Figure 3

### Resistors in Parallel

Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance. Each resistor thus has the full voltage of the source applied to it (Figure 1).

Each resistor draws the same current it would if it were the only resistor connected to the voltage source. This is true of the circuitry in a house or apartment. Each outlet that is connected to a appliance (the "resistor") can operate independently, and the current does not have to pass through each appliance sequentially.

### Ohm's Law and Parallel Resistors

Each resistor in the circuit has the full voltage (Figure 2). According to Ohm's law, the currents flowing through the individual resistors are $I_1 = \frac{V}{R_1}$, $I_2 = \frac{V}{R_2}$, and $I_3 = \frac{V}{R_3}$. Conservation of charge implies that the total current is the sum of these currents:

$I = I_1 + I_2 + I_3.$

Substituting the expressions for individual currents gives:

$I = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3}$

or

$I = V( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3})$

This implies that the total resistance in a parallel circuit is equal to the sum of the inverse of each individual resistances. Therefore, for every circuit with $n$ number or resistors connected in parallel,

$R_{n \;(parallel)} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} ... + \frac{1}{R_n}.$

This relationship results in a total resistance that is less than the smallest of the individual resistances. When resistors are connected in parallel, more current flows from the source than would flow for any of them individually, so the total resistance is lower.

Each resistor in parallel has the same full voltage of the source applied to it, but divide the total current amongst them. This is exemplified by connecting two light bulbs in a parallel circuit with a 1.5V battery. In a series circuit, the two light bulbs would be half as dim when connected to a single battery source. However, if the two light bulbs were connected in parallel, they would be equally as bright as if they were connected individually to the battery. Because the same full voltage is being applied to both light bulbs, the battery would also die more quickly, since it is essentially supplying full energy to both light bulbs. In a series circuit, the battery would last just as long as it would with a single light bulb, only the brightness is then divided amongst the bulbs.

#### Key Term Glossary

battery
A device that produces electricity by a chemical reaction between two substances.
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circuit
A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T
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conservation
A particular measurable property of an isolated physical system does not change as the system evolves.
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current
The time rate of flow of electric charge.
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energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
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inverse
Opposite in effect or nature or order.
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Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
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light
The natural medium emanating from the sun and other very hot sources (now recognised as electromagnetic radiation with a wavelength of 400-750 nm), within which vision is possible.
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ohm
in the International System of Units, the derived unit of electrical resistance; the electrical resistance of a device across which a potential difference of one volt causes a current of one ampere; symbol: Ω
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parallel
An arrangement of electrical components such that a current flows along two or more paths.
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resistance
The opposition to the passage of an electric current through that element.
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resistor
An electric component that transmits current in direct proportion to the voltage across it.
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series
A number of things that follow on one after the other or are connected one after the other.
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voltage
The amount of electrostatic potential between two points in space.