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medium difficulty10 min

Series and Parallel Combination of Resistors

A combination of resistors is an arrangement of two or more resistors connected in a circuit so that they can be replaced by a single equivalent resistance producing the same current-voltage effect between two terminals.

In one line: Series gives a longer path; parallel gives more paths.

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Understand it

  • In series, resistors are connected end to end and the same current passes through each resistor.
  • The applied potential difference divides across them.
  • In parallel, resistors are connected across the same two terminals, so the potential difference across each branch is the same and current divides among branches.
  • Equivalent resistance is larger than any individual resistor in series and smaller than the smallest branch resistance in parallel.

Example

Two resistors of 3 ohm and 6 ohm in series give 9 ohm. The same resistors in parallel give 2 ohm, which is smaller than both 3 ohm and 6 ohm.

Formula and rule

Series: Rs = R1 + R2 + R3 + ..., same current through all resistors, voltage divides.

Parallel: 1/Rp = 1/R1 + 1/R2 + 1/R3 + ..., same voltage across each branch, current divides.

SI unit of resistance is ohm.

Use series formula only when components carry the same current; use parallel formula only when components share the same two terminals.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find equivalent resistance of 4 ohm and 12 ohm in parallel, then in series with 2 ohm.

  1. 1Parallel part: 1/Rp = 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3, so Rp = 3 ohm.
  2. 2Total series resistance = 3 + 2 = 5 ohm.
  3. 3If connected to 10 V, current = 10/5 = 2 A.

Method you can reuse

  1. 1Redraw the circuit if needed.
  2. 2Identify resistors with the same current as series.
  3. 3Identify resistors connected between the same two nodes as parallel.
  4. 4Replace one simple group at a time by its equivalent resistance.
  5. 5Continue until a single resistance remains.
  6. 6Apply Ohm's law if source voltage is given.

Visual tip

Series diagram: resistors connected one after another in a single path. Parallel diagram: resistor branches connected between the same two junctions. Label current I in series and branch currents I1, I2 in parallel, with common voltage V across branches.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is equivalent resistance in a parallel combination less than the smallest resistor?

Think first, then show answer
In a parallel combination, each new branch provides an additional path for current. Since total current drawn from the source increases for the same voltage, the equivalent resistance decreases and becomes less than the smallest individual branch resistance.

Common wrong answer

For 4 ohm and 12 ohm in parallel, writing Rp = 16 ohm.

Why it loses marks: 16 ohm is the series sum, not the parallel equivalent.

Don't mix this up

Students sometimes add resistances directly in a parallel combination. Direct addition is valid only for series, not for parallel.

Exam tip

Before calculation, mark whether current is common or voltage is common. This usually reveals whether the connection is series or parallel.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask for equivalent resistance of mixed circuits, comparison of brightness of bulbs, current division, voltage division, or identifying series-parallel parts from a circuit diagram.
  1. 1

    Enough for 1 mark

    A combination of resistors is an arrangement of two or more resistors connected in a circuit so that they can be replaced by a single equivalent resistance producing the same current-voltage effect between two terminals.

  2. 2

    Add for 2 marks

    Example: Two resistors of 3 ohm and 6 ohm in series give 9 ohm. The same resistors in parallel give 2 ohm, which is smaller than both 3 ohm and 6 ohm.

  3. 3

    Add for 3 marks

    In series, resistors are connected end to end and the same current passes through each resistor.

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