derive, using Kirchhoff’s laws, a formula for the combined resistance of two or more resistors in series

Kirchhoff’s Laws – Derivations, Applications & Practical Skills (Cambridge AS & A Level Physics 9702)

1. Learning Outcomes (Cambridge Syllabus 10 D.C. circuits – 10.2)

• AO1 – Knowledge & Understanding: State KCL and KVL, apply the correct sign‑convention, and derive the series and parallel resistance formulas.
• AO2 – Application: Use the derived formulas to analyse potential dividers, include source internal resistance, and solve multi‑mesh circuits.
• AO3 – Practical Skills: Plan, carry out and evaluate an experiment to determine equivalent resistance, propagate uncertainties and discuss systematic errors.

2. Formal Statements & Sign Conventions

• Kirchhoff’s Current Law (KCL) – At any junction the algebraic sum of currents is zero:

  \$\displaystyle\sum I{\text{in}}-\sum I{\text{out}}=0\$

• Kirchhoff’s Voltage Law (KVL) – The algebraic sum of potential differences round any closed loop is zero:

  \$\displaystyle\sum V{\text{rise}}-\sum V{\text{drop}}=0\$

Sign convention (clockwise traversal taken as positive):

• Voltage rise (e.g., across a battery) → +
• Voltage drop (e.g., across a resistor) → –

Reversing the direction of traversal changes the sign of every term but does not affect the final result.

3. Derivation of Equivalent Resistance

3.1 Resistors in Series

1. Consider an ideal battery of emf \$V\$ (internal resistance neglected) driving \$n\$ resistors \$R1,R2,\dots,R_n\$ in series.
2. The same current \$I\$ flows through each element.
3. KVL (clockwise positive):

   \$V - I R1 - I R2 - \dots - I R_n = 0\$

4. Factor \$I\$:

   \$V = I\,(R1+R2+\dots+R_n)\$

5. Define the equivalent resistance \$R{\text{eq}}\$ by \$V = I R{\text{eq}}\$:

   \$\boxed{R{\text{eq(series)}} = \displaystyle\sum{k=1}^{n} R_k}\$

3.2 Resistors in Parallel

1. Connect the same \$n\$ resistors across an ideal battery of emf \$V\$.
2. Voltage across each resistor is the same: \$V_k = V\$.
3. KCL at the junction where the branch currents meet the main line:

   \$I = I1 + I2 + \dots + I_n\$

4. Ohm’s law for each branch: \$Ik = V/Rk\$.
5. Substitute into KCL:

   \$I = V\!\left(\frac{1}{R1}+\frac{1}{R2}+\dots+\frac{1}{R_n}\right)\$

6. Define \$R{\text{eq}}\$ by \$I = V/R{\text{eq}}\$:

   \$\boxed{\frac{1}{R{\text{eq(parallel)}}}= \displaystyle\sum{k=1}^{n}\frac{1}{R_k}}\$

3.3 Including Source Internal Resistance

If the battery has an internal resistance \$r\$, the loop equation becomes

\$V{\text{emf}} - I r - \sum{k=1}^{n} I R_k = 0\$

Hence the terminal voltage is

\$V{\text{term}} = V{\text{emf}} - I r = I\,R_{\text{eq}}\$

When \$r\$ is comparable to \$R_{\text{eq}}\$, the available voltage drops significantly – a point often examined in Paper 3/5 practical questions.

4. Application – Resistive Potential Divider

Two series resistors \$R1\$ and \$R2\$ are connected across a source \$V{\text{in}}\$. The output voltage \$V{\text{out}}\$ is taken across \$R_2\$.

1. Total resistance: \$R{\text{tot}} = R1+R_2\$.
2. Current through the series chain: \$I = \dfrac{V{\text{in}}}{R1+R_2}\$.
3. Drop across \$R_2\$ (the output):

   \$V{\text{out}} = I R2 = \frac{V{\text{in}}\,R2}{R1+R2}\$

Thus

\$\boxed{V{\text{out}} = V{\text{in}}\;\frac{R2}{R1+R_2}}\$

This relation is used extensively for biasing transistor circuits, sensor interfaces and voltage references.

5. Solving a Two‑Mesh Circuit (KCL + KVL)

Figure 1 shows a simple two‑mesh circuit with one battery \$V\$, three resistors \$R1,R2,R3\$, and mesh currents \$I1\$ (left loop) and \$I_2\$ (right loop).

V

R1

R2

R3

5.1 Writing the Equations

• KVL – Mesh 1 (clockwise, includes battery, \$R1\$, \$R2\$)

  \$V - I1R1 - I1R2 + I2R2 = 0\$

• KVL – Mesh 2 (clockwise, includes \$R2\$, \$R3\$)

  \$-I2R3 - I2R2 + I1R2 = 0\$

• KCL – at the node where the two meshes meet (implicitly satisfied by the mesh‑current formulation).

5.2 Solving the Simultaneous Equations

Arrange the equations in matrix form:

\[

\begin{bmatrix}

R1+R2 & -R_2\\[4pt]

-R2 & R2+R_3

\end{bmatrix}

\begin{bmatrix}

I1\\ I2

\end{bmatrix}

=

\begin{bmatrix}

V\\ 0

\end{bmatrix}

\]

Using Cramer’s rule or matrix inversion, the solutions are:

\[

I1 = \frac{V\,(R2+R3)}{(R1+R2)(R2+R3)-R2^{\,2}},\qquad

I2 = \frac{V\,R2}{(R1+R2)(R2+R3)-R_2^{\,2}}

\]

Both currents are positive, confirming that the assumed clockwise directions are correct. If a calculated current were negative, the actual direction would be opposite to the assumed one.

5.3 Physical Interpretation

• The denominator is the determinant of the resistance matrix – it represents the overall opposition to current flow in the coupled meshes.
• Increasing \$R1\$ reduces \$I1\$ more than \$I2\$, while increasing \$R3\$ mainly limits \$I_2\$.
• When \$R_2\to0\$ the two meshes become electrically shorted and the currents merge into a single loop.

6. Practical Activity – Determining Equivalent Resistance (AO3)

6.1 Assessment Objectives Addressed

6.2 Equipment

• DC power supply (adjustable, ≤ 12 V) with known internal resistance \$r\$ (or a battery with datasheet \$r\$).
• Resistor set (e.g., 10 Ω, 22 Ω, 47 Ω, tolerance ± 1 %).
• Digital multimeter (voltmeter/ammeter) – accuracy ± (0.5 % + 2 digits).
• Breadboard and short, flexible connecting leads.
• Four‑wire (Kelvin) leads for the resistor under test (optional – reduces lead‑resistance error).

6.3 Procedure

1. Build the series network of three resistors as shown in Fig 2. Record the colour code values.
2. Connect the ammeter in series and the voltmeter across the whole network (terminal voltage).
3. Set the supply to a convenient voltage (e.g., 6 V) and record the current \$Is\$ and voltage \$Vs\$.
4. Calculate the experimental equivalent resistance:

   \$R{\text{eq}}^{\; \text{exp}} = \frac{Vs}{I_s}\$

5. Repeat steps 1‑4 with the same three resistors in parallel.
6. For each configuration, repeat the measurement three times and take the mean values.
7. Propagate the uncertainties (see Section 6.5) and compute the percentage error relative to the theoretical value.
8. Discuss sources of systematic error (internal resistance of the supply, contact resistance, multimeter loading) and propose improvements.

6.4 Sample Data Table

6.5 Uncertainty Propagation for \$R = V/I\$

For independent uncertainties \$\Delta V\$ and \$\Delta I\$:

\[

\frac{\Delta R}{R}= \sqrt{\left(\frac{\Delta V}{V}\right)^{2}+\left(\frac{\Delta I}{I}\right)^{2}}

\]

Insert the measured \$\Delta V\$ and \$\Delta I\$ from the multimeter specifications to obtain \$\Delta R\$.

6.6 Suggested Improvements (AO3.5)

• Use a four‑wire (Kelvin) connection for the resistor under test – eliminates lead resistance from the measurement of \$V\$.
• Measure the internal resistance \$r\$ of the power supply (e.g., by the voltage‑drop method) and correct the terminal voltage accordingly.
• Employ a constant‑current source instead of a constant‑voltage source to avoid loading effects when the equivalent resistance changes.

7. Summary Table – Key Formulae & Uses

8. Common Misconceptions & How to Avoid Them

• Current changes in series: The same current flows through every series element; only the voltage drop varies.
• Mix‑up of series & parallel formulas: Remember “add resistances” → series; “add reciprocals” → parallel.
• Neglecting internal resistance: Always check the specification of the source; for high‑current loads the drop \$Ir\$ can be comparable to the external voltage.
• Sign errors in KVL: Choose a traversal direction, stick to it, and mark rises (+) and drops (–) consistently.
• Lead‑resistance error in measurements: Use four‑wire connections or subtract the measured lead resistance.

9. Extension Questions (Challenge for AO2)

1. Derive the equivalent resistance of a network where \$R1\$ and \$R2\$ are in parallel and this combination is in series with \$R_3\$. Hint: treat the parallel pair as a single resistor first.
2. Using the potential‑divider formula, calculate \$V{\text{out}}\$ for \$R1=4.7\;\text{k}\Omega\$, \$R2=10\;\text{k}\Omega\$, \$V{\text{in}}=12\;\text{V}\$.
3. For the two‑mesh circuit in Section 5, let \$V=9\;\text{V}\$, \$R1=10\;\Omega\$, \$R2=20\;\Omega\$, \$R3=30\;\Omega\$. Compute \$I1\$ and \$I_2\$ and comment on the direction of each current relative to the assumed clockwise arrows.

10. Cross‑Reference Checklist (Placement in a Full Unit)

This module should be embedded within a larger unit on “Electric Circuits & Fields”. Ensure the following connections are made:

• Before this module: Electromotive force, internal resistance, Ohm’s law, basic series/parallel circuits.
• After this module: AC circuits, reactance, impedance, power in AC, and electromagnetic induction (required for later syllabus sections).
• Practical links: The activity in Section 6 can be combined with the “Measuring internal resistance of a cell” practical (Paper 5) to reinforce the internal‑resistance concept.

11. Module Status

Standalone Module – Covers all required outcomes for 10.2 Kirchhoff’s laws. It should be taught together with the surrounding topics listed in the checklist to provide a coherent progression through the Cambridge AS & A Level Physics syllabus.