6. Series and Parallel Combination of Resistors
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In practical circuits, multiple resistors (or resistive components such as bulbs) are commonly connected together in one of two basic arrangements: series or parallel. Understanding the rules governing these combinations, and being able to apply the resulting formulas, is one of the most frequently tested numerical skills in this chapter.
Series Combination
In a series combination, resistors are connected end to end, one after another, so that there is only a single path for current to flow through all of them. Because there is only one path, the same current flows through every resistor in the series combination. The total (equivalent) resistance of resistors in series is simply the sum of their individual resistances:
R(series) = R₁ + R₂ + R₃ + ...
The equivalent resistance in a series combination is always greater than the greatest individual resistance in the combination. The total potential difference across the combination is divided among the individual resistors in proportion to their resistance (a larger resistor gets a larger share of the voltage) — this is known as the principle of potential divider.
Parallel Combination
In a parallel combination, resistors are connected between the same two points, so that there are multiple separate paths for current to flow, and each resistor has the same potential difference applied across it. The current from the source divides among the branches in inverse proportion to their resistance (more current flows through the path of lower resistance). The reciprocal of the equivalent resistance of resistors in parallel is the sum of the reciprocals of the individual resistances:
1 / R(parallel) = 1/R₁ + 1/R₂ + 1/R₃ + ...
The equivalent resistance in a parallel combination is always smaller than the smallest individual resistance in the combination — adding more parallel paths always makes it easier, not harder, for current to flow overall. For exactly two resistors in parallel, a convenient shortcut formula is Rparallel = (R₁ × R₂) / (R₁ + R₂). Household electrical circuits are wired in parallel, not series, and this is an extremely important exam fact with real practical reasoning behind it, discussed further in the table below.
Feature | Series Combination | Parallel Combination |
|---|---|---|
Current | Same current flows through every component | Current divides among the branches; unequal in general |
Voltage | Divides among components in proportion to resistance | Same voltage across every component |
Equivalent resistance | Sum of all resistances (increases); greater than the largest individual R | Reciprocal sum (decreases); smaller than the smallest individual R |
Effect of one component failing | Entire circuit breaks (all components stop working) | Other components continue to work normally |
Use in household wiring | Not used, for the reason above | Standard method — each appliance can be switched independently and rated at the same 220–240 V supply |
Total power | Sum of individual powers (I same in each, P = I²R) | Sum of individual powers (V same across each, P = V²/R) |