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Measurement of Electrical Resistance of Brass Specimen Using Carey Foster Bridge

A precise bridge method for low-resistance measurement, eliminating end corrections by interchanging the gaps

Physics Experiments 20 September, 2026 10 min read

Measurement of Electrical Resistance of Brass Specimen Using Carey Foster Bridge

A precise bridge method for low-resistance measurement, eliminating end corrections by interchanging the gaps

Aim

To measure the electrical resistance of a given brass specimen (wire coil or strip) using the Carey Foster Bridge method, and to understand how this modified Wheatstone Bridge technique eliminates end corrections for accurate low-resistance measurement.

Apparatus Required

Carey Foster Bridge
Brass specimen (coil or strip)
Standard resistance box (0.1–10 Ω)
Galvanometer (sensitive)
Jockey (sharp-tipped)
Leclanché cell (1.5 V)
Thick copper strips (2 nos.)
Connecting wires
Plug key / tapping key
Note: Thick copper strips serve as short-circuiting links with negligible resistance. They are used to calibrate the bridge wire by replacing X and Y with near-zero resistances during the determination of ρ.

Theory & Principle

The Carey Foster Bridge

The Carey Foster Bridge is a modified form of the Wheatstone Bridge specifically designed for comparing two nearly equal resistances with high precision. While a standard metre bridge uses only four arms, the Carey Foster Bridge introduces two additional resistances (P and Q) connected in series with the bridge wire, making it sensitive to very small resistance differences.

In the standard Wheatstone Bridge, balance is achieved when the ratio condition is satisfied:

P / Q = R / S
Wheatstone Bridge balance condition — derived from Kirchhoff's current and voltage laws

The Carey Foster Bridge extends this principle. An unknown resistance X is placed in the left gap, and a known standard resistance Y is placed in the right gap. The bridge wire of length L (typically 100 cm) and uniform cross-section connects the two gaps. A galvanometer is connected between the junction of P and Q and a jockey that slides along the bridge wire.

Derivation of the Working Formula

Let α and β represent the end corrections at the left and right ends of the bridge wire respectively, and let ρ be the resistance per unit length of the bridge wire.

Configuration 1 — X in the left gap, Y in the right gap. The balance point is found at distance l1 from the left end. At balance:

P / Q = [X + α + l1ρ] / [Y + β + (L − l1)ρ]

Configuration 2 — X and Y are interchanged. The new balance point is at l2. At balance:

P / Q = [Y + α + l2ρ] / [X + β + (L − l2)ρ]

Since P and Q are unchanged, the right-hand sides of both equations are equal. Cross-multiplying and simplifying, the end corrections α and β cancel out, yielding the Carey Foster Bridge formula:

X − Y = (l1 − l2) ρ
End corrections are completely eliminated by interchanging the gaps

To use this formula, we first need to determine ρ using two known resistances (or two thick copper strips with a small known resistance in Y).

Why Brass Has Higher Resistance Than Copper

Brass is an alloy of copper (roughly 60–70%) and zinc (30–40%). When zinc atoms substitute into the copper crystal lattice, they disrupt the regular periodic arrangement that allows conduction electrons to move freely. These substitutional impurity atoms act as scattering centres, increasing the frequency of electron collisions and thereby raising the electrical resistivity. This behaviour is described quantitatively by Nordheim's rule, which states that the resistivity of a binary alloy increases with the product x(1 − x), where x is the fraction of the solute element.

Resistivity comparison:
Copper: ρCu ≈ 1.7 × 10−8 Ω·m
Brass: ρbrass ≈ 6 × 10−8 Ω·m (roughly 3.5× higher)

This significant difference means that a brass wire of the same dimensions as a copper wire will have approximately 3.5 times the resistance, making it an ideal specimen for this experiment since the resistance falls in a measurable range on the Carey Foster Bridge.

Connection to Kirchhoff's Laws

The Wheatstone Bridge balance condition, which underlies the Carey Foster Bridge, is derived from Kirchhoff's laws:

  • Kirchhoff's Current Law (KCL): At the galvanometer node, the current through the galvanometer is zero at balance, meaning the current entering equals the current leaving through the bridge arms alone.
  • Kirchhoff's Voltage Law (KVL): Around the loop containing the galvanometer, the algebraic sum of potential differences is zero. This gives the ratio condition P/Q = R/S when Ig = 0.

The Carey Foster Bridge simply distributes part of the arm resistance along a calibrated wire, allowing a continuous null adjustment instead of discrete resistance steps.

Circuit Diagram

0 50 100 cm Bridge Wire (length L) X (Brass) Left gap Y (R-box) Right gap P Resistance Q Resistance G Jockey Key (K) E (Cell) − + I l (balance point)

Figure: Carey Foster Bridge circuit. The brass specimen X is in the left gap and the standard resistance box Y is in the right gap. The galvanometer is connected between the junction of P and Q and the jockey sliding on the bridge wire.

Procedure

Part A: Determination of ρ (Resistance per Unit Length of Bridge Wire)

  1. Place thick copper strips in both the left and right gaps of the Carey Foster Bridge. These have negligible resistance and effectively short-circuit both gaps.
  2. Introduce a small known resistance Y0 (say 0.5 Ω) from the standard resistance box into the right gap alongside the copper strip. Close the key and find the balance point l1 by sliding the jockey along the wire until the galvanometer shows zero deflection.
  3. Interchange the copper strip and the standard resistance: place Y0 in the left gap and the copper strip in the right gap. Find the new balance point l2.
  4. Calculate ρ using the formula: ρ = Y0 / (l1 − l2). Repeat with different values of Y0 and take the mean ρ.

Part B: Measurement of Brass Specimen Resistance X

  1. Remove the copper strips. Place the brass specimen in the left gap and a known resistance Y from the resistance box in the right gap.
  2. Close the key and gently slide the jockey along the bridge wire to locate the balance point where the galvanometer reads zero. Record this distance as l1 from the left end.
  3. Interchange the gaps: move the brass specimen to the right gap and the resistance box to the left gap. Find the new balance point l2.
  4. Calculate X using: X = Y + (l1 − l2) ρ
  5. Repeat the measurement for at least five different values of Y (0.5, 1.0, 1.5, 2.0, 2.5 Ω) and record l1 and l2 each time.
  6. Calculate X for each observation and take the mean value as the final result.
Precautions:
  • The jockey should be pressed gently on the wire to avoid damaging it or altering its cross-section.
  • All connections must be tight and clean to minimise contact resistance.
  • The battery key should be closed before the galvanometer key, and opened in reverse order, to protect the galvanometer from transient surges.
  • Ensure P and Q are equal (typically 2 Ω each) to keep the bridge sensitivity high near the centre of the wire.
  • The bridge wire should be straight, taut, and free of kinks.
  • Allow the cell to stabilise before taking readings; avoid prolonged current flow to prevent heating of the wire.

Observation Tables

Table A: Determination of ρ (Resistance per Unit Length of Bridge Wire)

Using thick copper strips in gaps. P = Q = 2 Ω.

S.No. Y0 (Ω) l1 (cm)
Y0 in right gap		 l2 (cm)
Y0 in left gap		 l1 − l2 (cm) ρ = Y0/(l1−l2)
(Ω/cm)
10.555.644.511.10.04505
21.061.038.822.20.04505
31.566.833.533.30.04505
Mean ρ = (0.04505 + 0.04505 + 0.04505) / 3 = 0.04505 Ω/cm
The consistent values of ρ across different Y0 confirm that the bridge wire is uniform.

Table B: Measurement of Brass Specimen Resistance X

Brass specimen in one gap, standard resistance Y in the other. P = Q = 2 Ω. ρ = 0.04505 Ω/cm.

S.No. Y (Ω) l1 (cm)
Brass left gap		 l2 (cm)
Brass right gap		 l1 − l2 (cm) X = Y + (l1−l2)ρ
(Ω)
10.568.431.237.20.5 + 1.676 = 2.18
21.063.236.626.61.0 + 1.198 = 2.20
31.557.642.015.61.5 + 0.703 = 2.20
42.052.847.25.62.0 + 0.252 = 2.25
52.547.052.8−5.82.5 − 0.261 = 2.24

Note: In observation 5, Y > X, so l1 < l2 and the difference is negative. The formula X = Y + (l1 − l2)ρ still holds — the negative product correctly reduces Y toward X. As Y approaches X, the balance point nears the centre of the wire (50 cm mark).

Calculations

Sample Calculation (Observation 2):

Given: Y = 1.0 Ω,  l1 = 63.2 cm,  l2 = 36.6 cm,  ρ = 0.04505 Ω/cm

Using the Carey Foster Bridge formula:
X = Y + (l1 − l2) × ρ
X = 1.0 + (63.2 − 36.6) × 0.04505
X = 1.0 + 26.6 × 0.04505
X = 1.0 + 1.198
X = 2.198 Ω ≈ 2.20 Ω

Mean Value

Mean X = (2.18 + 2.20 + 2.20 + 2.25 + 2.24) / 5
= 11.07 / 5
= 2.214 Ω

Rounding to appropriate significant figures: X ≈ 2.18 Ω (taking the median of the five values as a robust estimator)

The spread of individual readings (2.18 to 2.25 Ω) gives an estimate of the measurement uncertainty. The standard deviation of the five values is approximately 0.03 Ω, leading to a standard error of 0.03/√5 ≈ 0.013 Ω. Accounting for systematic uncertainties in the determination of ρ and contact resistance variations, we report an overall uncertainty of ±0.08 Ω.

Result

The electrical resistance of the given brass specimen, measured using the Carey Foster Bridge method, is:
X = 2.18 ± 0.08 Ω

This value is consistent with the expected resistance of a brass coil of the given dimensions. The higher resistance compared to copper (a similar copper specimen would yield approximately 0.6 Ω) confirms the effect of alloying on electrical conductivity. The Carey Foster Bridge method successfully eliminated end corrections through the interchange technique, as evidenced by the consistency of X values across different choices of Y.

Sources of Error

  1. Non-uniformity of the bridge wire: If the cross-sectional area of the bridge wire varies along its length, ρ is not constant and the assumed linear relationship between balance length and resistance breaks down. This is typically the most significant systematic error in this experiment.
  2. Contact resistance at the gaps: Loose or oxidised connections at the gaps where X and Y are placed introduce additional unknown resistances. Although the interchanging technique cancels the fixed end corrections α and β, any change in contact resistance between the two configurations introduces an error that is not cancelled.
  3. Temperature variation: Prolonged passage of current heats both the bridge wire and the brass specimen, changing their resistances during the experiment. The temperature coefficient of resistance for brass is approximately 0.001 per degree Celsius, so even a 1°C rise can shift X by about 0.002 Ω.
  4. Galvanometer sensitivity: If the galvanometer is insufficiently sensitive, the balance point can only be located to within 1–2 mm, introducing uncertainties of the order of (1 mm × ρ) ≈ 0.0005 Ω per reading, which accumulates across multiple readings.
  5. Thermoelectric EMFs: Junctions of dissimilar metals (brass-to-copper at the gaps, copper-to-solder at connections) can generate small thermoelectric voltages of the order of microvolts. These parasitic EMFs disturb the null condition. Reversing the battery polarity and averaging the two balance lengths for each configuration can help mitigate this error.

Viva Voce Questions

Q1. Why does brass have a higher electrical resistance than copper?

Brass is an alloy of copper and zinc, typically containing 30–40% zinc by weight. When zinc atoms occupy sites in the copper crystal lattice, they disrupt the regular periodic potential that allows conduction electrons to propagate with minimal scattering. Each zinc atom acts as a scattering centre, increasing the collision frequency of the conduction electrons and thereby reducing their mean free path. According to Nordheim's rule for substitutional binary alloys, the residual resistivity increases proportionally to x(1 − x), where x is the solute (zinc) fraction. As a result, the resistivity of brass is approximately 6 × 10−8 Ω·m — about 3.5 times higher than pure copper at 1.7 × 10−8 Ω·m.

Q2. What is the Carey Foster Bridge formula?

The Carey Foster Bridge formula is: X − Y = (l1 − l2) ρ, where X is the unknown resistance, Y is the known standard resistance from the resistance box, l1 is the balance length with X in the left gap, l2 is the balance length after interchanging X and Y between the two gaps, and ρ is the resistance per unit length of the bridge wire. The quantity ρ is determined separately by placing known resistances (or thick copper strips) in the gaps and applying the same interchange procedure. This formula is powerful because the end corrections at both ends of the bridge wire cancel out completely in the subtraction.

Q3. How does interchanging the gaps eliminate end corrections?

End corrections arise because the effective length of the bridge wire differs slightly from the measured length due to contact resistances at the points where the wire meets the gap blocks. Let α and β be the equivalent lengths of these end corrections at the left and right ends respectively. In configuration 1 (X left, Y right), the balance condition gives: P(Y + β + (L − l1)ρ) = Q(X + α + l1ρ). In configuration 2 (X right, Y left), it gives: P(X + β + (L − l2)ρ) = Q(Y + α + l2ρ). When these two equations are divided (or subtracted after rearranging), the terms containing α and β appear identically on both sides and cancel, leaving the clean result X − Y = (l1 − l2)ρ. This elimination is the principal advantage of the Carey Foster method over a simple metre bridge.

Q4. What happens if the bridge wire is non-uniform?

If the bridge wire has a non-uniform cross-section — for example, it is thinner at one end than the other — then the resistance per unit length ρ varies along the wire. In that case, the formula X − Y = (l1 − l2)ρ is no longer valid because ρ is not a constant. The balance points will still exist, but the relationship between measured length differences and resistance differences becomes nonlinear. This introduces systematic errors that cannot be removed by the interchange technique. To detect non-uniformity, one can determine ρ at several positions along the wire by using different pairs of known resistances and checking whether ρ remains constant. If significant non-uniformity is found, a calibration curve mapping cumulative resistance to length must be constructed and used instead of a single ρ value.

Q5. How is this experiment related to the Wheatstone Bridge?

The Carey Foster Bridge is a direct modification of the Wheatstone Bridge. In a standard Wheatstone Bridge, four discrete resistances P, Q, R, and S form a closed quadrilateral, with a galvanometer bridging two opposite junctions. At balance, no current flows through the galvanometer, and the condition P/Q = R/S holds — a result derived by applying Kirchhoff's current law at the nodes and Kirchhoff's voltage law around the loops. In the Carey Foster Bridge, two of the arms (equivalent to R and S) are replaced by the combination of gap resistances (X and Y) with segments of a calibrated resistance wire. The continuous wire allows infinitely fine adjustment of the balance point by sliding the jockey, unlike the discrete steps of a resistance box. The fundamental null-balance principle is identical in both instruments; the Carey Foster design simply optimises the measurement for comparing two nearly equal low resistances by reading small length differences, and it adds the gap-interchange technique to eliminate end corrections that would limit accuracy in a simple metre bridge.

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