understand the use of a galvanometer in null methods

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{R2}=Vs\frac{R2}{R1+R2}\$

• 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.

1. KCL (junction rule) at the central node

   \$\sum I{\text{into}} = \sum I{\text{out}} \;\;\Longrightarrow\;\; I1 = I2 \qquad\text{(1)}\$

   At null the galvanometer draws negligible current, so the same current flows in the left and right branches.

2. KVL (loop rule) for the left loop (supply → R₁ → R₂ → back to supply)

   \$Vs - I1(R1+R2)=0 \;\;\Longrightarrow\;\; I1=\frac{Vs}{R1+R2} \qquad\text{(2)}\$

3. KVL for the right loop (supply → R_adj → R_ref → back to supply)

   \$Vs - I2(R{\text{adj}}+R{\text{ref}})=0 \;\;\Longrightarrow\;\; I2=\frac{Vs}{R{\text{adj}}+R{\text{ref}}} \qquad\text{(3)}\$

4. Equating the currents from (1)–(3) gives the null‑condition relationship

   \$\frac{R2}{R1+R2}= \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.

1. Write the null‑condition from (4):

   \$\frac{R2}{R1+R_2}= \frac{3.0}{3.0+1.0}=0.75\$

2. Cross‑multiply:

   \$R2 =0.75(R1+R_2)\$

3. Collect terms:

   \$R2-0.75R2 =0.75R1 \;\;\Longrightarrow\;\;0.25R2 =0.75R_1\$

4. Solve for R₂:

   \$R2 =3R1 =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.

1. Connect the potentiometer wire of total resistance R_p across a stable supply V_s (the reference voltage).
2. Place a sliding contact (the jockey) at a distance l from one end; the resistance up to the jockey is

   \$Rl =\frac{l}{L}\,Rp\$

   where L is the total length of the wire.

3. Connect the unknown emf E_u in series with the galvanometer and the short segment of wire at the jockey.
4. Move the jockey until the galvanometer shows zero deflection. At null:

   \$\frac{Eu}{Vs}= \frac{Rl}{Rp}= \frac{l}{L} \qquad\text{(5)}\$

5. Hence the unknown emf is obtained from the calibrated ratio:

   \$Eu = Vs\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

8. Planning Template (AO3 – Experimental Skills)

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{R2}{R1+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):

\$Eu = Vs\frac{l}{L}=6.00\ \text{V}\times\frac{0.75}{1.50}=3.00\ \text{V}\$

11. Sources of Error and Mitigation

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.