Recall and use in calculations, the fact that: (a) the sum of the currents entering a junction in a parallel circuit is equal to the sum of the currents that leave the junction (b) the total p.d. across the components in a series circuit is equal to

4.3.2 Series and Parallel Circuits

Learning objectives

• State and apply the junction (current) rule: the algebraic sum of currents entering a node equals the sum leaving it.
• State and apply the series‑current rule: the same current flows through every component of a series circuit.
• State and apply the series potential‑difference (p.d.) rule: the total p.d. across components in series equals the sum of the individual p.d.s.
• State and apply the parallel p.d. rule: all branches of a parallel arrangement have the same p.d. as the whole arrangement.
• Calculate equivalent resistance for series and parallel networks.
• Calculate the power dissipated in each resistor.
• (Extended syllabus) Use Kirchhoff’s loop rule for circuits containing several sources or mixed series‑parallel arrangements.

Standard circuit symbols (Cambridge IGCSE)

Safety reminder (practical work)

• Check battery polarity before connecting the circuit.
• Never connect a voltmeter across a live wire that is not part of a complete circuit – this can cause a short circuit.
• Use insulated wires and keep hands away from exposed metal parts.
• Turn off the power supply before rearranging components.

1. Junction (current) rule – parallel circuits

At any node the algebraic sum of currents is zero:

\[

\sum I{\text{in}} = \sum I{\text{out}}

\]

If a current \(I\) enters a junction and splits into two branches, the rule becomes

\[

I = I{1}+I{2}\qquad(\text{units: A})

\]

This follows directly from the conservation of charge.

2. Series‑current rule

In a series arrangement the same current flows through every component:

\[

I = I{1}=I{2}=I_{3}= \dots \qquad(\text{units: A})

\]

Consequently, the current measured anywhere in the series chain is the total current supplied by the source.

3. Series potential‑difference rule

When resistors are connected end‑to‑end, the total p.d. supplied by the source equals the sum of the individual voltage drops:

\[

V{\text{total}} = V{1}+V{2}+V{3}+ \dots \qquad(\text{units: V})

\]

Using Ohm’s law (\(V = IR\)) for each resistor:

\[

V{\text{total}} = I R{1}+ I R{2}+ I R{3}+ \dots = I\,(R{1}+R{2}+R_{3}+ \dots)

\]

Hence the equivalent resistance of resistors in series is

\[

R{\text{total}} = R{1}+R{2}+R{3}+ \dots \qquad(\Omega)

\]

4. Parallel potential‑difference rule

All branches of a parallel network experience the same p.d. as the whole network:

\[

V{\text{across parallel}} = V{\text{branch 1}} = V_{\text{branch 2}} = \dots \qquad(\text{V})

\]

5. Equivalent resistance

• Series: \(R{\text{eq}} = \displaystyle\sum R{i}\)
• Parallel: \(\displaystyle\frac{1}{R{\text{eq}}}= \sum\frac{1}{R{i}}\)  or \(R{\text{eq}} = \dfrac{1}{\displaystyle\sum\frac{1}{R{i}}}\)

6. Power in a resistor

\[

P = I V = I^{2}R = \frac{V^{2}}{R}\qquad(\text{W})

\]

Any of the three forms may be used depending on the quantities that are known.

7. Optional – Kirchhoff’s loop rule (extended syllabus)

For any closed loop the algebraic sum of the potential differences is zero:

\[

\sum V{\text{rise}} - \sum V{\text{drop}} = 0 \qquad(\text{V})

\]

This is a general statement of the series p.d. rule and is required only for the extended part of the syllabus.

8. Worked examples

Example 1 – Parallel circuit: verification of the junction rule

• Battery emf: \(V = 12\;\text{V}\)
• Resistances: \(R{1}=4\;\Omega,\;R{2}=6\;\Omega,\;R_{3}=12\;\Omega\)

1. Branch currents (using \(I = V/R\))

   • \(I_{1}=12/4 = 3.0\;\text{A}\)
   • \(I_{2}=12/6 = 2.0\;\text{A}\)
   • \(I_{3}=12/12 = 1.0\;\text{A}\)

2. Total current supplied by the battery

   \[

   I{\text{total}} = I{1}+I{2}+I{3}=3.0+2.0+1.0=6.0\;\text{A}

   \]

3. Junction rule check – the 6.0 A entering the node equals the sum of the three currents leaving it, confirming the rule.
4. Equivalent resistance

   \[

   \frac{1}{R_{\text{eq}}}= \frac{1}{4}+\frac{1}{6}+\frac{1}{12}=0.500\;\Omega^{-1}

   \quad\Rightarrow\quad

   R_{\text{eq}} = 2.0\;\Omega

   \]

5. Power dissipated in each resistor

   \[

   P{1}=I{1}^{2}R_{1}=3.0^{2}\times4=36\;\text{W},\;

   P_{2}=2.0^{2}\times6=24\;\text{W},\;

   P_{3}=1.0^{2}\times12=12\;\text{W}

   \]

   Total power \(P_{\text{total}} = 36+24+12 = 72\;\text{W}\), which also equals \(V I = 12 \times 6 = 72\;\text{W}\).

Example 2 – Series circuit: verification of the series p.d. rule

• Current through the series chain: \(I = 0.5\;\text{A}\)
• Resistances: \(R{1}=10\;\Omega,\;R{2}=20\;\Omega,\;R_{3}=30\;\Omega\)

1. Voltage drop across each resistor (using \(V = I R\))

   \[

   V_{1}=0.5\times10=5.0\;\text{V},\;

   V_{2}=0.5\times20=10.0\;\text{V},\;

   V_{3}=0.5\times30=15.0\;\text{V}

   \]

2. Total p.d. supplied

   \[

   V{\text{total}} = V{1}+V{2}+V{3}=5.0+10.0+15.0=30.0\;\text{V}

   \]

3. Series‑current rule check – the same 0.5 A flows through each resistor.
4. Equivalent resistance

   \[

   R_{\text{eq}} = 10+20+30 = 60\;\Omega

   \quad\text{and}\quad

   V{\text{total}} = I R{\text{eq}} = 0.5\times60 = 30.0\;\text{V}

   \]

5. Power dissipated in each resistor

   \[

   P{1}=I^{2}R{1}=0.5^{2}\times10=2.5\;\text{W},\;

   P_{2}=0.5^{2}\times20=5.0\;\text{W},\;

   P_{3}=0.5^{2}\times30=7.5\;\text{W}

   \]

   Total power \(= 2.5+5.0+7.5 = 15.0\;\text{W}\), which also equals \(V_{\text{total}} I = 30.0 \times 0.5 = 15.0\;\text{W}\).

9. Summary of key relations

10. Suggested classroom investigation (AO3)

1. Build a parallel network on a breadboard using three resistors (e.g., 4 Ω, 6 Ω, 12 Ω) and a 12 V battery.
2. Measure the current in each branch with a digital multimeter (ammeter placed in series with the branch).
3. Record the total current supplied by the battery.
4. Compare the measured values with the calculated currents from Example 1. Discuss any differences (instrument tolerance, wire resistance, contact resistance, etc.).
5. Re‑arrange the same three resistors in series, measure the single current, and verify the series‑current and series p.d. rules using Example 2 as a guide.

11. Quick reference diagram (for revision)

Insert a labelled diagram showing:

• A battery feeding a parallel network of three resistors with currents \(I{1}, I{2}, I_{3}\) and total current \(I\).
• A separate diagram of a series chain of three resistors with voltage drops \(V{1}, V{2}, V_{3}\) and the same current \(I\) throughout.