Cambridge Syllabus Notes

Potential Dividers – Use of a Galvanometer in Null‑Method Measurements

Objective

To understand how a galvanometer is employed in null‑method measurements with potential dividers and potentiometers, and to be able to plan, set up, analyse and evaluate such experiments to the standard required for Cambridge International AS & A Level Physics (9702).

1. The Basic Potential Divider

Two resistors, R₁ and R₂, are connected in series across a known supply voltage V_s.
The voltage across R₂ is given by the divider formula $$V_{R_2}=V_s\frac{R_2}{R_1+R_2}$$
By varying the ratio R₂/(R₁+R₂) any desired fraction of V_s can be obtained.

2. Null‑Method Measurements and the Galvanometer

A null method compares two potentials until the galvanometer reads zero (no current).
Because the galvanometer current is essentially zero at null, the circuit is not loaded – the measured quantity is not altered.
Typical sensitivity of a laboratory galvanometer: 10 µA ÷ 10 µV (depends on the instrument).

3. Kirchhoff’s Laws Applied to a Wheatstone Bridge

Consider the Wheatstone‑type circuit shown in the figure (see below). The galvanometer (G) connects the two junctions.

KCL (junction rule) at the central node $$\sum I_{\text{into}} = \sum I_{\text{out}} \;\;\Longrightarrow\;\; I_1 = I_2 \qquad\text{(1)}$$ At null the galvanometer draws negligible current, so the same current flows in the left and right branches.
KVL (loop rule) for the left loop** (supply → R₁ → R₂ → back to supply)** $$V_s - I_1(R_1+R_2)=0 \;\;\Longrightarrow\;\; I_1=\frac{V_s}{R_1+R_2} \qquad\text{(2)}$$

KVL for the right loop** (supply → R_adj → R_ref → back to supply)** $$V_s - I_2(R_{\text{adj}}+R_{\text{ref}})=0 \;\;\Longrightarrow\;\; I_2=\frac{V_s}{R_{\text{adj}}+R_{\text{ref}}} \qquad\text{(3)}$$

Equating the currents from (1)–(3) gives the null‑condition relationship $$\frac{R_2}{R_1+R_2}= \frac{R_{\text{adj}}}{R_{\text{adj}}+R_{\text{ref}}} \qquad\text{(4)}$$

This derivation satisfies the syllabus requirement to “derive and apply Kirchhoff’s laws”.

Worked Example – Determining an Unknown Resistance

Given: R₁=2.0 kΩ, R_adj=3.0 kΩ, R_ref=1.0 kΩ. Find the unknown resistor R₂ when the galvanometer is at null.

Write the null‑condition from (4): $$\frac{R_2}{R_1+R_2}= \frac{3.0}{3.0+1.0}=0.75$$

Cross‑multiply: $$R_2 =0.75(R_1+R_2)$$

Collect terms: $$R_2-0.75R_2 =0.75R_1 \;\;\Longrightarrow\;\;0.25R_2 =0.75R_1$$

Solve for R₂: $$R_2 =3R_1 =3\times2.0\ \text{kΩ}=6.0\ \text{kΩ}$$

4. The Potentiometer – A Null‑Method Voltage Comparator

Definition: A potentiometer is a uniform resistance wire (or calibrated resistor bank) used as a potential divider to compare an unknown emf with a known reference voltage. No current is drawn from the source being measured when the null condition is achieved.

Connect the potentiometer wire of total resistance R_p across a stable supply V_s (the reference voltage).

Place a sliding contact (the jockey) at a distance l from one end; the resistance up to the jockey is $$R_l =\frac{l}{L}\,R_p$$ where L is the total length of the wire.

Connect the unknown emf E_u in series with the galvanometer and the short segment of wire at the jockey.

Move the jockey until the galvanometer shows zero deflection. At null: $$\frac{E_u}{V_s}= \frac{R_l}{R_p}= \frac{l}{L} \qquad\text{(5)}$$

Hence the unknown emf is obtained from the calibrated ratio: $$E_u = V_s\frac{l}{L}$$

5. emf, Terminal Potential Difference and Internal Resistance

e.m.f. (E) – the work done per coulomb by a source when no current flows (open‑circuit condition).

Terminal potential difference (V) – the voltage actually measured across the source terminals when a current I flows.

Internal resistance (r) – the resistance inherent to the source. The relationship is $$V = E - I r \qquad\text{(6)}$$

In a null‑method measurement the galvanometer current is essentially zero (I≈0), so from (6) the terminal voltage equals the emf of the source being compared. This is the reason why null methods give higher accuracy than direct‑reading voltmeters.

6. Role of the Galvanometer in Null Methods

Detects the point at which the potential difference between two nodes is exactly zero.

Because the current through it at null is negligible, the circuit is not loaded.

Provides a very sensitive indication of the balance point (often to a few µV).

7. Advantages of the Null Method

Benefit Explanation

Minimal loading The galvanometer draws only a tiny current at null, so the measured quantity is not altered.

High accuracy Zero‑deflection can be detected to a fraction of the galvanometer’s full‑scale deflection, giving resolutions of 10⁻⁴ – 10⁻⁵ of the full scale.

Elimination of systematic errors No direct current or voltage is read, so errors due to instrument internal resistance are avoided.

8. Planning Template (AO3 – Experimental Skills)

Section What to Fill In

Aim State the quantity to be measured (e.g., “To determine the emf of an unknown cell using a potentiometer.”)

Apparatus & Materials List all items with specifications (e.g., potentiometer wire 1.5 m, 150 Ω; regulated 6.00 V supply; galvanometer 10 µA ÷ 10 µV; jockey, connecting leads, etc.).

Safety & Precautions Identify hazards (e.g., hot wires, battery reversal), and note precautions (use insulated leads, avoid short‑circuits, keep the circuit off while changing connections).

Method (step‑by‑step) Provide a concise numbered procedure that can be copied into the exam answer sheet.

Data Table Prepare a table for recording length l, supply voltage, and any known resistor values.

Calculations Show the formulae to be used (e.g., (5) for emf, (4) for unknown resistance) and indicate where to substitute measured values.

Evaluation List possible sources of error and how they will be assessed (e.g., repeatability, percentage uncertainty).

9. Experimental Procedure Checklist (During the Experiment)

Zero the galvanometer with its terminals shorted; re‑check after any major adjustment.

Verify that the supply voltage is stable (measure with a multimeter).

Check all resistor values (colour code or multimeter) and note tolerances.

Ensure all connections are clean, tight, and, where possible, use four‑wire (Kelvin) leads.

Connect the circuit exactly as shown in the schematic (see below).

Adjust the variable resistor (or move the jockey) slowly; watch for the first sign of galvanometer deflection reversal.

When the galvanometer reads zero, record:

Length l (or the value of the adjustable resistor).

Supply voltage V_s (if not already known to high accuracy).

All known resistor values.

Repeat the measurement at least two more times, interchanging the positions of the unknown and reference sources if applicable, to assess reproducibility.

10. Example Calculations

10.1 Determining an Unknown Resistance (Wheatstone Bridge)

Data: R₁=2.0 kΩ, R_adj=3.0 kΩ, R_ref=1.0 kΩ, galvanometer at null.

Using equation (4): $$\frac{R_2}{R_1+R_2}= \frac{3.0}{3.0+1.0}=0.75$$ Solving gives R₂=6.0 kΩ (see Worked Example above).

10.2 Determining an Unknown emf Using a Potentiometer

Data: Potentiometer length L = 1.50 m, total resistance R_p = 150 Ω, supply V_s = 6.00 V, null at l=0.75 m.

From (5): $$E_u = V_s\frac{l}{L}=6.00\ \text{V}\times\frac{0.75}{1.50}=3.00\ \text{V}$$

11. Sources of Error and Mitigation

Error Source Effect on Measurement Mitigation

Galvanometer zero drift Apparent null point shifted → systematic offset Zero the galvanometer with its terminals shorted before each trial; re‑check after large adjustments.

Thermal EMF at contacts Spurious voltages (µV–mV) add to the measured potential Use identical metals for connections, keep junctions symmetric, and allow the circuit to reach thermal equilibrium.

Contact resistance / dirty terminals Alters effective resistance ratios, especially in a Wheatstone bridge Clean contacts, tighten screws, or employ four‑wire (Kelvin) connections where feasible.

Supply voltage fluctuation Changes divider ratios during measurement Use a regulated DC source or a fresh high‑capacity battery; monitor with a voltmeter.

Inaccurate length measurement on a potentiometer Directly affects the calculated emf Use a calibrated scale, read the jockey position at eye level, and repeat measurements.

12. Schematic Diagram

13. Summary Checklist (Before & During the Experiment)

Zero the galvanometer.

Confirm a stable supply voltage.

Verify all known resistor values and tolerances.

Ensure clean, tight connections (use four‑wire leads if possible).

Set up the circuit exactly as in the schematic.

Adjust the variable element slowly; note the first zero‑deflection point.

Record all required data (length or resistance, supply voltage, known resistances).

Calculate the unknown quantity using the appropriate null‑condition equation.

Repeat the measurement to assess reproducibility and estimate uncertainty.

14. Further Reading

– Cambridge International A‑Level Physics (9702), Chapter 10 “Electrical Measurements”.
– J. D. Jackson, Classical Electrodynamics, sections on precision measurement techniques.
– J. R. D. Taylor, Principles of Measurement, Chapter 4 on null methods and potentiometers.

understand the use of a galvanometer in null methods

Potential Dividers – Use of a Galvanometer in Null‑Method Measurements

Objective

1. The Basic Potential Divider

2. Null‑Method Measurements and the Galvanometer

3. Kirchhoff’s Laws Applied to a Wheatstone Bridge

Worked Example – Determining an Unknown Resistance

4. The Potentiometer – A Null‑Method Voltage Comparator

5. emf, Terminal Potential Difference and Internal Resistance

6. Role of the Galvanometer in Null Methods

7. Advantages of the Null Method

8. Planning Template (AO3 – Experimental Skills)

9. Experimental Procedure Checklist (During the Experiment)

10. Example Calculations

10.1 Determining an Unknown Resistance (Wheatstone Bridge)

10.2 Determining an Unknown emf Using a Potentiometer

11. Sources of Error and Mitigation

12. Schematic Diagram

13. Summary Checklist (Before & During the Experiment)

14. Further Reading

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Benefit	Explanation
Minimal loading	The galvanometer draws only a tiny current at null, so the measured quantity is not altered.
High accuracy	Zero‑deflection can be detected to a fraction of the galvanometer’s full‑scale deflection, giving resolutions of 10⁻⁴ – 10⁻⁵ of the full scale.
Elimination of systematic errors	No direct current or voltage is read, so errors due to instrument internal resistance are avoided.

Section	What to Fill In
Aim	State the quantity to be measured (e.g., “To determine the emf of an unknown cell using a potentiometer.”)
Apparatus & Materials	List all items with specifications (e.g., potentiometer wire 1.5 m, 150 Ω; regulated 6.00 V supply; galvanometer 10 µA ÷ 10 µV; jockey, connecting leads, etc.).
Safety & Precautions	Identify hazards (e.g., hot wires, battery reversal), and note precautions (use insulated leads, avoid short‑circuits, keep the circuit off while changing connections).
Method (step‑by‑step)	Provide a concise numbered procedure that can be copied into the exam answer sheet.
Data Table	Prepare a table for recording length l, supply voltage, and any known resistor values.
Calculations	Show the formulae to be used (e.g., (5) for emf, (4) for unknown resistance) and indicate where to substitute measured values.
Evaluation	List possible sources of error and how they will be assessed (e.g., repeatability, percentage uncertainty).

Error Source	Effect on Measurement	Mitigation
Galvanometer zero drift	Apparent null point shifted → systematic offset	Zero the galvanometer with its terminals shorted before each trial; re‑check after large adjustments.
Thermal EMF at contacts	Spurious voltages (µV–mV) add to the measured potential	Use identical metals for connections, keep junctions symmetric, and allow the circuit to reach thermal equilibrium.
Contact resistance / dirty terminals	Alters effective resistance ratios, especially in a Wheatstone bridge	Clean contacts, tighten screws, or employ four‑wire (Kelvin) connections where feasible.
Supply voltage fluctuation	Changes divider ratios during measurement	Use a regulated DC source or a fresh high‑capacity battery; monitor with a voltmeter.
Inaccurate length measurement on a potentiometer	Directly affects the calculated emf	Use a calibrated scale, read the jockey position at eye level, and repeat measurements.