Physics · Ch 3 — Current Electricity

Grouping of Resistances: Series, Parallel and Mixed

3.8

Grouping of Resistances: Series, Parallel and Mixed

Series combination. When resistors $R_1, R_2, R_3, \ldots$ are connected end-to-end in a single chain, so that exactly the same current $I$ must flow through each one in turn (there being no other path available), the combination is said to be in series. Applying $V = IR$ to each resistor separately, the potential difference across the whole combination is the sum of the individual drops:

V = V_1 + V_2 + V_3 + \cdots = I R_1 + I R_2 + I R_3 + \cdots = I(R_1+R_2+R_3+\cdots)

so that the single equivalent resistance $R_s = V/I$ of a series combination is simply

R_s = R_1 + R_2 + R_3 + \cdots

always LARGER than even the largest individual resistance present, since every extra resistor in series can only add further to the total obstruction to current flow.

Parallel combination. When several resistors are instead connected between the SAME pair of nodes, so that the SAME potential difference $V$ appears across every one of them, but the total current $I$ supplied by the source is free to split up and take whichever branch(es) it can, the combination is said to be in parallel. Applying $I = V/R$ to each branch separately, the total current is the sum of the individual branch currents:

I = I_1 + I_2 + I_3 + \cdots = \frac{V}{R_1} + \frac{V}{R_2} + \frac{V}{R_3} + \cdots = V\left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\cdots\right)

so that the equivalent resistance $R_p = V/I$ of a parallel combination satisfies

\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \cdots

always SMALLER than even the smallest individual resistance present, since adding an extra parallel branch always opens up an additional path for current, never restricting the paths already available. For just two resistors in parallel, this reduces to the frequently useful special case $R_p = R_1R_2/(R_1+R_2)$ . …

Figure 1Series and parallel combination of resistors

What this figure shows. Two small circuit diagrams are drawn side by side. The LEFT diagram shows three resistors $R_1$ , $R_2$ , $R_3$ drawn as standard zig-zag resistor symbols connected one after another in a single chain (the right end of $R_1$ wired directly to the left end of $R_2$ , and the right end of $R_2$ wired directly to the left end of $R_3$ ), with the free left end of $R_1$ and the free right end of $R_3$ each connected by wire to opposite terminals of a battery, and an ammeter shown in the main wire measuring the SAME current $I$ that flows through all three resistors in turn -- illustrating a SERIES combination. The RIGHT diagram shows the same three resistors $R_1$ , $R_2$ , $R_3$ instead drawn as three separate parallel branches connected between a shared left node and a shared right node, so that all three resistors' left ends connect to the same left node and all three right ends connect to the same right node, with the two nodes connected to a battery, and separate ammeters shown in each b …

Table 2Series versus parallel combination of resistors: formula comparison

	Series combination	Parallel combination
Equivalent resistance	$R_s = R_1 + R_2 + \cdots$ (always larger than the largest $R_i$ )	$1/R_p = 1/R_1 + 1/R_2 + \cdots$ (always smaller than the smallest $R_i$ )
Current through each resistor	Same through every resistor, $I$	Different through each; splits in inverse proportion to $R_i$
Potential difference across each	Different across each; splits in direct proportion to $R_i$	Same across every resistor, $V$

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