recall and use the circuit symbols shown in section 6 of this syllabus

Practical Circuits – Objective

Recall and use the circuit symbols shown in Section 6 of the Cambridge International AS & A Level Physics (9702) syllabus, and apply them confidently to the practical investigations required for the 2025‑2027 syllabus.

1. Circuit Symbols (Section 6)

2. Conventions for Drawing Practical Circuit Diagrams

1. Place the battery (or source) on the left; the positive terminal is at the top.
2. Draw wires as straight lines; use a small open circle to indicate a junction.
3. Connect an ammeter in series with the load so the same current passes through it.
4. Connect a voltmeter in parallel across the component whose voltage is required.
5. Show switches where the circuit can be opened/closed; label the switch state (open/closed).
6. Label every component with its nominal value (e.g. \$R=10\;\Omega\$, \$C=100\;\mu\text{F}\$).
7. Indicate the direction of conventional current with an arrow on the wire or on diodes/LEDs.
8. Use the correct symbol for a galvanometer or potentiometer when required.

3. Essential Theory (Sections 9‑10)

3.1 Kirchhoff’s Laws

• First Law (Current Law): The algebraic sum of currents entering a junction equals the sum leaving it.

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

• Second Law (Voltage Law): The algebraic sum of the potential differences around any closed loop is zero.

  \$\sum V = 0\$

Worked example – series‑parallel network

A 12 V battery with internal resistance \$r=0.5\;\Omega\$ supplies a parallel combination of \$R1=4\;\Omega\$ and \$R2=6\;\Omega\$. Using the two laws:

1. Find the equivalent resistance: \$R_{\text{eq}} = \left(\frac{1}{4}+\frac{1}{6}\right)^{-1}=2.4\;\Omega\$.
2. Total resistance \$RT = r + R{\text{eq}} = 2.9\;\Omega\$.
3. Total current \$I = \dfrac{E}{R_T}= \dfrac{12}{2.9}=4.14\;\text{A}\$.
4. Current in each branch: \$I1 = \dfrac{V{\text{ab}}}{R1}\$, \$I2 = \dfrac{V{\text{ab}}}{R2}\$ where \$V_{\text{ab}} = E - Ir = 12-4.14\times0.5=9.93\;\text{V}\$.
5. Thus \$I1=2.48\;\text{A}\$ and \$I2=1.66\;\text{A}\$.

3.2 Series‑Parallel Resistor Networks

• Series: \$R{\text{s}} = R1+R_2+\dots\$
• Parallel: \$\displaystyle\frac{1}{R{\text{p}}}= \frac{1}{R1}+ \frac{1}{R_2}+ \dots\$
• Combine series and parallel steps to simplify any network before applying Kirchhoff’s laws.

3.3 Internal Resistance of a Cell

Model a real cell as an ideal emf \$E\$ in series with an internal resistance \$r\$.

\[

I = \frac{E}{R+r},\qquad V_{\text{terminal}} = E - Ir

\]

Experimental method: vary the external resistance \$R\$, record \$I\$ and \$V\$, then plot \$V\$ against \$I\$. The straight‑line fit gives

• Slope \$= -r\$ (internal resistance)
• Intercept \$= E\$ (emf)

3.4 Potential Divider

Two series resistors \$R1\$ and \$R2\$ produce a fraction of the source voltage:

\[

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

\]

Commonly used with a rheostat to obtain an adjustable voltage for sensors, LEDs or to bias a transistor.

3.5 Wheatstone Bridge (Null‑Method)

Four resistors form a diamond shape. A galvanometer \$G\$ connects the two mid‑points, and a battery supplies the bridge.

• Balance condition (no current through \$G\$):\[

  \frac{R1}{R2}= \frac{R3}{Rx}

  \]

• When the bridge is balanced, \$R_x\$ (the unknown resistance) can be found from the known values.
• Practical use: precise resistance measurement, calibration of \$R_x\$, or as a stepping‑stone to the potentiometer.

3.6 Potentiometer / Null‑Method Technique

A uniform resistance wire of length \$L\$ carries a known current, producing a linear potential gradient.

1. Connect the unknown emf \$E_{\text{u}}\$ in series with a galvanometer to a sliding contact on the wire.
2. Adjust the contact until the galvanometer reads zero (null point).
3. Read the length \$l\$ from the zero‑point to the left end; the known voltage \$V{\text{ref}}\$ across the whole wire gives the gradient \$V{\text{ref}}/L\$.
4. Then \$E{\text{u}} = V{\text{ref}}\;\dfrac{l}{L}\$.

This technique is required for Paper 5 (practical) questions.

3.7 Thevenin and Norton Equivalent Circuits

• Thevenin equivalent: a single voltage source \$E{\text{Th}}\$ in series with a resistance \$R{\text{Th}}\$ that reproduces the external behaviour of a more complex network.
• Norton equivalent: a single current source \$I{\text{N}}\$ in parallel with a resistance \$R{\text{N}}\$ (where \$R{\text{N}} = R{\text{Th}}\$).
• Useful for analysing circuits that contain multiple sources and loads, especially in AO2 questions.

4. Practical Investigations (AO3 – Planning, Execution, Evaluation)

4.1 Determining the Internal Resistance of a Cell

1. Assemble the circuit using the symbols from Section 1.
2. Select at least five values of \$R\$ (e.g. 5 Ω, 10 Ω, 20 Ω, 50 Ω, 100 Ω).
3. For each \$R\$, record the current \$I\$ (ammeter) and the terminal voltage \$V\$ (voltmeter). Use the appropriate ranges to minimise instrument error.
4. Plot \$V\$ (y‑axis) against \$I\$ (x‑axis). Fit a straight line \$V = -rI + E\$.
5. Read the gradient (gives \$-r\$) and the y‑intercept (gives \$E\$). Estimate uncertainties from the scatter of points.
6. Evaluation prompt: Discuss sources of error (contact resistance, instrument loading, temperature change) and suggest improvements (four‑wire measurement, use of a potentiometer to reduce loading).

4.2 Wheatstone Bridge Balance

1. Set up the bridge with three known resistors and one unknown \$R_x\$.
2. Adjust the variable resistor (or sliding contact) until the galvanometer reads zero.
3. Record the values of the three known resistors and calculate \$R_x\$ using the balance condition.
4. Repeat with different combinations to check consistency.
5. Safety & Uncertainty checklist:

   • Use low voltage (≤ 6 V) to avoid heating the resistors.
   • Choose ammeter/galvanometer range that gives a readable deflection without saturating.
   • Estimate random error from repeated readings (± 0.1 Ω typical) and systematic error from the tolerance of the known resistors.

6. Evaluation prompt: Comment on the effect of contact resistance at the bridge junctions and how a four‑wire technique could improve accuracy.

4.3 Potentiometer Measurement of an Unknown emf

1. Connect a uniform resistance wire of length \$L\$ to a stable DC source; measure the total voltage \$V_{\text{ref}}\$ across the wire.
2. Place the unknown emf source in series with a galvanometer and a sliding contact on the wire.
3. Slide the contact until the galvanometer reads zero (null point). Record the length \$l\$ from the left end to the contact.
4. Calculate the unknown emf: \$E{\text{u}} = V{\text{ref}}\dfrac{l}{L}\$.
5. Repeat with different \$V_{\text{ref}}\$ values to verify linearity and estimate uncertainties.

5. Safety & Uncertainty Checklist (AO3)

• Instrument selection: Choose the smallest range that gives a full‑scale deflection without exceeding the instrument limits.
• Connection order: Connect the ammeter (or galvanometer) last, after the circuit is powered, to avoid accidental short‑circuits.
• Power off before modifying: Always disconnect the source before adding/removing components or changing the switch state.
• Temperature effects: Allow components to reach thermal equilibrium; note any drift in readings.
• Uncertainty estimation:

  • Random error: take at least three readings for each setting and calculate the standard deviation.
  • Systematic error: consider instrument tolerance, lead resistance, and internal resistance of the source.

• Documentation: Record circuit diagram, component values, instrument ranges, raw data, and calculated results in a tidy table.

6. Cross‑Topic Applications (Linking to Other Syllabus Areas)

• Particle physics (Section 13): Detector circuits (e.g., Geiger‑Müller tube) use a high‑voltage battery symbol, a resistor for bias, and a counting circuit (galvanometer/amperemeter).
• Thermodynamics & Energy (Section 15): RC charging curves are analysed to determine time constants, linking electrical energy storage to thermal processes.
• Oscillations (Section 16): LC resonance circuits employ the inductor and capacitor symbols; the voltage across the capacitor is displayed on an oscilloscope (voltmeter with “~”).
• Quantum & Nuclear (Section 22): Photodiode or photovoltaic cell circuits use the diode symbol, a load resistor, and a voltmeter to measure photocurrent.

7. Further Applications – Extending to Later Topics (Sections 16‑22)

8. Practice Questions

1. Wheatstone Bridge Diagram – List the symbols required to draw a bridge that measures an unknown resistance \$R_x\$. Include: battery, four resistors (two known, one unknown, one variable), a galvanometer, and a switch.
2. Numerical – Internal Resistance – A 12 V battery has \$r = 0.5\;\Omega\$. It supplies a \$4\;\Omega\$ resistor, an ammeter \$A\$, and a voltmeter \$V\$ as shown in the diagram below.

   • Calculate the current shown by the ammeter.
   • Calculate the voltage read by the voltmeter (terminal voltage).
   • Show all steps using the symbols from the table.

3. Conceptual – Explain why a voltmeter must be connected in parallel and an ammeter in series, referring to the function of each symbol and the effect on the circuit.
4. Diagram in Words – Describe (in words) a circuit that powers a 9 V battery, a current‑limiting resistor, and an LED. Indicate the direction of conventional current with arrows on the battery and LED symbols.
5. Wheatstone Bridge Calculation – In a balanced bridge \$R1 = 100\;\Omega\$, \$R2 = 150\;\Omega\$, \$R3 = 200\;\Omega\$. Find the unknown resistance \$Rx\$.
6. Thevenin Equivalent – A circuit consists of a 6 V battery in series with \$R1 = 2\;\Omega\$, and this series combination is in parallel with \$R2 = 4\;\Omega\$. Determine the Thevenin equivalent voltage and resistance seen by a load connected across the parallel combination.

9. Summary

• Master the Section 6 symbols – they are the language of every practical physics circuit.
• Apply Kirchhoff’s first and second laws, series‑parallel reduction, internal‑resistance analysis, potential divider, Wheatstone bridge, potentiometer, and Thevenin/Norton concepts to solve AO2 problems.
• Plan, execute, and evaluate experiments using the safety & uncertainty checklist; always record clear diagrams with correct symbols and labelled values.
• Link circuit work to other syllabus topics (particle physics, thermodynamics, oscillations, quantum) to demonstrate the interdisciplinary nature of practical physics.
• Regularly practice the questions above – they reinforce symbol recall, circuit drawing, quantitative analysis, and the critical evaluation skills required for the Cambridge AS & A Level Physics examinations.