5.2. Heating Effect of an Electric Current

Resistances in Series and Parallel

Once you’ve got a handle on Ohm’s Law, the next step is understanding how resistors combine in circuits. This is crucial, because in real circuits, you’ll often have a mix of series and parallel connections.

Resistances in Series

When resistors are connected one after another (in a straight line), they are in series.

The total resistance is simply the sum of the individual resistances:

Rtotal = R₁ + R₂ + R₃ + …

Key points:

• The current is the same through each resistor.

• The voltage across each resistor will be different (depending on its resistance).

Example
Two resistors, 5 Ω and 10 Ω, in series:
Rtotal = 5 + 10 = 15 Ω

Resistances in Parallel

When resistors are connected side by side (each branch connected to the same two points), they are in parallel.

The total resistance is given by:

1 / Rtotal = 1 / R₁ + 1 / R₂ + 1 / R₃ + …

Key points:

• The voltage across each resistor is the same.

• The current splits between the branches.

• The total resistance is always less than the smallest resistor in the group.

Example
Two resistors, 6 Ω and 12 Ω, in parallel:
1 / Rtotal = 1 / 6 + 1 / 12 = 2 / 12 + 1 / 12 = 3 / 12
So, Rtotal = 12 / 3 = 4 Ω

Mixed Circuits (Series + Parallel)

Sometimes circuits will have a combination of series and parallel connections. The trick is to simplify the circuit step by step.

1. Identify and combine any purely series or purely parallel sections.

2. Redraw the circuit after each simplification.

3. Repeat until you have a single total resistance.

4. Use Ohm’s Law (V = IR) to find current, voltage, or resistance as needed.

Example

• A 4 Ω and 6 Ω resistor are connected in parallel.

• This parallel combination is then connected in series with a 10 Ω resistor.

Step 1: Find the parallel section:
1 / Rtotal = 1 / 4 + 1 / 6 = 3 / 12 + 2 / 12 = 5 / 12
Rtotal = 12 / 5 = 2.4 Ω

Step 2: Add the series resistor:
Rtotal = Rparallel  + Rseries = 2.4 + 10 = 12.4 Ω

Tips for Problem Solving

• Always label voltages and currents in your diagram before starting calculations.

• Simplify the circuit gradually — don’t try to do it all at once.

• Double-check your answers:

  • For series, total resistance increases.

  • For parallel, total resistance decreases.