Explain that the sum of the currents into a junction is the same as the sum of the currents out of the junction

4.3.2 Series and Parallel Circuits – Core Syllabus Objectives

1. Kirchhoff’s Current Law (KCL) – Current‑Sum Rule

At a junction (node) the algebraic sum of currents is zero:

\[

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

\]

• Choose a sign convention (e.g. currents entering the node are positive, leaving are negative) and keep it consistent.
• KCL follows directly from the conservation of electric charge – charge cannot accumulate at a node.

2. Series Circuits

• Current behaviour: only one path exists, so the same current flows through every component.
  

  \[

  I{\text{series}} = I1 = I_2 = \dots

  \]

• Combined e.m.f. of cells (cells end‑to‑end):
  

  \[

  E{\text{total}} = E1 + E_2 + \dots

  \]

• Total resistance:
  

  \[

  R{\text{total}} = R1 + R_2 + \dots

  \]

• Voltage‑sum rule (KVL): the algebraic sum of potential differences around any closed series loop is zero.
  

  \[

  \sum V{\text{rise}} - \sum V{\text{drop}} = 0 \quad\text{or}\quad \sum V = 0

  \]

Illustrative Series Example

A 12 V battery supplies three resistors in series: \(R1=4\;\Omega\), \(R2=6\;\Omega\), \(R_3=2\;\Omega\).

• Total resistance: \(R_{\text{total}} = 4+6+2 = 12\;\Omega\).
• Current through the circuit: \(I = \dfrac{V}{R_{\text{total}}}= \dfrac{12\;\text{V}}{12\;\Omega}=1\;\text{A}\).
• Voltage across each resistor: \(V1=IR1=4\;\text{V}\), \(V2=6\;\text{V}\), \(V3=2\;\text{V}\).
  

  Check the voltage‑sum rule: \(4+6+2 = 12\;\text{V}\) (equals the source voltage).

3. Parallel Circuits

• Current behaviour: the total current from the source splits into the branches.
  

  \[

  I{\text{total}} = I1 + I_2 + \dots

  \]

  This is a direct application of KCL at the junction where the branches diverge.

• Voltage across each branch is the same as the source voltage (or the voltage between the common nodes).
• Total resistance:
  

  \[

  \frac{1}{R{\text{total}}}= \frac{1}{R1}+ \frac{1}{R_2}+ \dots

  \]

  Consequently, \(R_{\text{total}}\) is always less than the smallest individual resistance.

• Practical advantage of parallel wiring:

  • If one appliance fails (opens), the others continue to operate because each has its own path to the source.
  • Every appliance receives the full supply voltage, so each works at its rated brightness or power.

Parallel Example with KCL

A 9 V battery feeds three resistors in parallel: \(R1=3\;\Omega\), \(R2=6\;\Omega\), \(R_3=9\;\Omega\).

1. Equivalent resistance:

   \[

   \frac{1}{R_{\text{total}}}= \frac{1}{3}+\frac{1}{6}+\frac{1}{9}= \frac{6+3+2}{18}= \frac{11}{18}

   \qquad\Rightarrow\qquad

   R_{\text{total}}= \frac{18}{11}\approx1.64\;\Omega

   \]

2. Total current from the battery:

   \[

   I{\text{total}}= \frac{V}{R{\text{total}}}= \frac{9}{1.64}\approx5.5\;\text{A}

   \]

3. Branch currents (Ohm’s law, \(I=V/R\)):

   \[

   I_1=\frac{9}{3}=3\;\text{A},\qquad

   I_2=\frac{9}{6}=1.5\;\text{A},\qquad

   I_3=\frac{9}{9}=1\;\text{A}

   \]

4. Check KCL:

   \[

   I{\text{total}} = I1+I2+I3 = 3+1.5+1 = 5.5\;\text{A}

   \]

4. Numerical Practice – Applying KCL & KVL

5. Step‑by‑Step Problem‑Solving Strategy

1. Identify every node (junction) in the circuit diagram.
2. Assign a direction to each current (conventionally away from the positive terminal of the source).
3. Write a KCL equation** for each node using \(\sum I{\text{in}} = \sum I{\text{out}}\).
4. Apply Ohm’s law** (\(V = IR\)) to each branch to relate currents and voltages.
5. Use the series and parallel resistance formulas** to find equivalent resistances where required.
6. If a closed loop is involved, write a KVL (voltage‑sum) equation**: \(\sum V{\text{rise}} = \sum V{\text{drop}}\) (or \(\sum V = 0\) with sign convention).
7. Solve the simultaneous equations** algebraically to obtain the unknown currents, voltages or resistances.

6. Common Mistakes & How to Avoid Them

• Mixing signs in KCL: decide once whether entering currents are positive (or negative) and keep that convention throughout the problem.
• Assuming equal currents in parallel branches: only series branches share the same current; parallel branches generally have different currents.
• Forgetting the voltage‑sum rule in series: the algebraic sum of all potential differences around a closed series loop must be zero.
• Neglecting the reduction of resistance in parallel: adding another parallel path always lowers the equivalent resistance.
• Overlooking the practical advantage of parallel wiring: a single open circuit does not affect the remaining branches, and each branch receives the full supply voltage.

7. Key Formula Summary

8. Summary

Kirchhoff’s Current Law guarantees charge conservation at every node, giving the current‑sum rule for parallel circuits. Combined with the voltage‑sum rule (KVL), Ohm’s law, and the series/parallel resistance formulas, students can analyse any IGCSE‑level circuit. The practical advantage of wiring appliances in parallel—each receives the full supply voltage and remains operational if another branch fails—is explained directly by KCL. Mastery of these principles enables accurate calculation of currents, voltages, combined e.m.f., and equivalent resistances, fulfilling all core syllabus objectives for series and parallel circuits.