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)

Symbol	Name / Function	Typical Use in Experiments
⎓⎓	Ideal battery (emf $E$)	Provides a constant potential difference.
⎓⎓ ⎯ $r$	Battery with internal resistance $r$	Real cell; internal resistance shown inside the symbol.
—⎯⎯—	Resistor $R$	Controls current; value written next to the symbol.
—⎯⎯— ~	Variable resistor / rheostat	Adjustable resistance; often used in potential‑divider circuits.
—⎯⎯— ↦	Diode (forward direction shown by arrow)	Allows current in one direction only.
—⎯⎯— ↦↤	LED (light‑emitting diode)	Visual indicator of current flow.
||	Capacitor $C$	Stores charge; used in RC and AC circuits.
⟳	Inductor $L$	Produces a magnetic field; used in LC and AC circuits.
⏚	Switch (open/closed)	Opens or closes the circuit safely.
V	Voltmeter (connected in parallel)	Measures potential difference.
A	Ammeter (connected in series)	Measures current.
Ω	Ohmmeter (connected across the component being measured)	Measures resistance.
G	Galvanometer (sensitive ammeter)	Used in null‑method and bridge circuits.
⎓⎓ ~	AC source (function generator)	Provides sinusoidal voltage for AC, LC and resonance work.
—⎯⎯— ⚖	Potentiometer (sliding contact on a uniform wire)	Null‑balance measurement of emf or unknown voltage.

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 $R_1=4\;\Omega$ and $R_2=6\;\Omega$. 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 $R_T = r + R_{\text{eq}} = 2.9\;\Omega$.
3. Total current $I = \dfrac{E}{R_T}= \dfrac{12}{2.9}=4.14\;\text{A}$.
4. Current in each branch: $I_1 = \dfrac{V_{\text{ab}}}{R_1}$, $I_2 = \dfrac{V_{\text{ab}}}{R_2}$ where $V_{\text{ab}} = E - Ir = 12-4.14\times0.5=9.93\;\text{V}$.
5. Thus $I_1=2.48\;\text{A}$ and $I_2=1.66\;\text{A}$.

3.2 Series‑Parallel Resistor Networks

• Series: $R_{\text{s}} = R_1+R_2+\dots$
• Parallel: $\displaystyle\frac{1}{R_{\text{p}}}= \frac{1}{R_1}+ \frac{1}{R_2}+ \dots$
• 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 $R_1$ and $R_2$ produce a fraction of the source voltage:

\[ V_{\text{out}} = V_{\text{in}}\;\frac{R_2}{R_1+R_2} \]

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 $G$):\[ \frac{R_1}{R_2}= \frac{R_3}{R_x} \]
• 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)

Later Topic	Typical Practical Circuit	Key Symbols Used
RC charging/discharging (Thermodynamics & Energy)	Battery → switch → resistor $R$ → capacitor $C$ → back to battery; voltmeter across $C$.	Battery, switch, resistor, capacitor, voltmeter.
LC resonance (Oscillations)	AC source → inductor $L$ → capacitor $C$ → back to source; oscilloscope (voltmeter “~”) across $C$.	AC source, inductor, capacitor, voltmeter (AC).
AC RLC circuits (Electric & Magnetic Fields)	AC source → resistor $R$ → inductor $L$ → capacitor $C$ → back to source; ammeter in series, voltmeter in parallel with each component.	AC source, resistor, inductor, capacitor, ammeter, voltmeter.
Photoelectric effect (Quantum)	Photodiode (diode symbol) → load resistor → voltmeter; bias supplied by a small DC battery.	Diode, resistor, battery, voltmeter.
Thevenin/Norton analysis (Advanced circuit theory)	Complex network reduced to a single source + series/parallel resistance; symbols for source, resistors, and load.	Battery (or AC source), resistors, load resistor.

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 $R_1 = 100\;\Omega$, $R_2 = 150\;\Omega$, $R_3 = 200\;\Omega$. Find the unknown resistance $R_x$.
6. Thevenin Equivalent – A circuit consists of a 6 V battery in series with $R_1 = 2\;\Omega$, and this series combination is in parallel with $R_2 = 4\;\Omega$. 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.