Most circuits have more than one component, called a resistor that limits the flow of charge in the circuit. A measure of this limit on charge flow is called resistance. The simplest combinations of resistors are the series and parallel connections illustrated in Figure 1. The total resistance of a combination of resistors depends on both their individual values and how they are connected.

These energies must be equal, because there is no other source and no other destination for energy in the circuit. Thus, qV = qV1 + qV2 + qV3. The charge q cancels, yielding V = V1 + V2 + V3, as stated. (Note that the same amount of charge passes through the battery and each resistor in a given amount of time, since there is no capacitance to store charge, there is no place for charge to leak, and charge is conserved.) Now substituting the values for the individual voltages gives

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V = IR1 + IR2 + IR3 = I(R1 + R2 + R3).

Note that for the equivalent single series resistance Rs, we have

V = IRs.

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This implies that the total or equivalent series resistance Rs of three resistors is Rs = R1 + R2 + R3. This logic is valid in general for any number of resistors in series; thus, the total resistance Rs of a series connection is

Rs = R1 + R2 + R3+…,

as proposed. Since all of the current must pass through each resistor, it experiences the resistance of each, and resistances in series simply add up.

Section Summary

• The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual resistances: Rs = R1 + R2 + R3 +….
• Each resistor in a series circuit has the same amount of current flowing through it.
• The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input.
• The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be determined using the formula:
• Each resistor in a parallel circuit has the same full voltage of the source applied to it.
• The current flowing through each resistor in a parallel circuit is different, depending on the resistance.
• If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.

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