Physics · Electricity

Solving series and parallel circuits

You can add series resistors, but the parallel part always gives you the wrong total.

In a series circuit the current is shared, and in a parallel circuit the voltage is shared. Use that one idea to decide what to add and what to keep.

This lesson is part of SPM Physics electricity. It uses the laws from applying potential difference, resistance and Ohm’s law.

What are the combination rules?

Arrangement	Total resistance	Current	Voltage
Series	R = R₁ + R₂ + …	same through each	adds to the supply
Parallel	1/R = 1/R₁ + 1/R₂ + …	adds to the total	same across each branch

Worked example: a mixed circuit

A 12 V battery of negligible internal resistance is connected to a 5 Ω resistor in series with a parallel pair of 4 Ω and 12 Ω resistors.

Pass 1: reduce. For the parallel pair, 1/R = 1/4 + 1/12 = 3/12 + 1/12 = 4/12, so R = 3 Ω. The total is 5 + 3 = 8 Ω.

Pass 2: expand. Total current I = 12 ÷ 8 = 1.5 A. Voltage across the 5 Ω resistor is 1.5 × 5 = 7.5 V. The pair then has 12 − 7.5 = 4.5 V.

Branch currents: 4.5 ÷ 4 = 1.125 A and 4.5 ÷ 12 = 0.375 A. Check: 1.125 + 0.375 = 1.5 A, which matches the total.

The mistake that loses marks

A common slip is to add 1/4 and 1/12 to get 1/3 and then write the parallel resistance as 1/3 Ω. The last step, inverting, was skipped.

Always finish by inverting. If the answer is larger than the smallest resistor, the parallel total is wrong.

Check yourself

A 9.0 V supply is connected to a 4 Ω resistor in series with 6 Ω and 3 Ω resistors in parallel. Find the current from the supply and the voltage across the parallel pair.

Answer

Parallel pair: 1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6, so R = 2 Ω. Total resistance = 4 + 2 = 6 Ω.

Current I = 9.0 ÷ 6 = 1.5 A. Voltage across the 4 Ω resistor is 1.5 × 4 = 6.0 V, so the parallel pair has 9.0 − 6.0 = 3.0 V.

What to study next

Move on to calculating electrical energy and power. Then test the skill with the electricity practice set.

The simple circuit calculation tutor lets you check each pass. For a teacher to work through circuits with you, see online one-to-one Physics tuition or the one-hour trial class (from RM50).

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Common questions

Why is the parallel resistance smaller than either resistor?

Each parallel branch gives the current another path, so more charge flows for the same voltage. More current for the same voltage means a smaller resistance. The total is always smaller than the smallest branch.

What stays the same in series and what stays the same in parallel?

In series the current is the same through every component, and the voltages add to the supply. In parallel the voltage across every branch is the same, and the currents add to the total current.

How do I handle a mixed circuit?

Reduce it first: replace each parallel group with one equivalent resistor, then add the series parts. Find the total current. Then expand back, using the voltage across the group to get each branch current.

Do I need the reciprocal formula for two resistors?

For two resistors, R = (R₁ × R₂) ÷ (R₁ + R₂) is quicker. For three or more, use the reciprocals, and invert the result at the end. Both give the same answer.

Keep going

When a teacher works one-to-one with your circuit diagrams, they can watch which pass of the method you skip, and ask you to explain each current before you calculate it.

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