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Measurement of Electrical Resistance of Copper Wire Using Carey Foster Bridge

Determine the unknown resistance of a copper wire specimen and the resistance per unit length of the bridge wire using the null-point interchange method.

Physics Experiments 20 September, 2026 10 min read
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Carey Foster Bridge

Measurement of Electrical Resistance of Copper Wire Using Carey Foster Bridge

Determine the unknown resistance of a copper wire specimen and the resistance per unit length of the bridge wire using the null-point interchange method.

Subject: Physics Level: B.Sc. / Class 12 Duration: ~45 min

Aim

To measure the electrical resistance of a given copper wire using the Carey Foster Bridge and to determine the resistance per unit length (ρ) of the bridge wire.

Apparatus Required

  • Carey Foster Bridge (with a 1 m uniform manganin or constantan slide wire mounted on a metre scale)
  • Copper wire specimen (unknown resistance X)
  • Standard resistance box (range 0.1–10 Ω)
  • Galvanometer (centre-zero type, high sensitivity)
  • Jockey (sharp-tipped sliding contact)
  • Leclanché cell or battery eliminator (2 V DC)
  • Two thick copper strips (for short-circuiting gaps during calibration)
  • Connecting wires (insulated, low resistance)
  • One-way key (plug key / tapping key)

Theory & Principle

What is a Carey Foster Bridge?

The Carey Foster Bridge is a modified form of the Wheatstone Bridge specifically designed to measure small resistances with high precision. It consists of four resistances arranged in a bridge configuration: two fixed ratio arms P and Q, and two gaps where the unknown resistance X and a known resistance Y (from a resistance box) are connected. A uniform slide wire of length 100 cm bridges the two gaps, and a galvanometer connected between the central junction and a sliding jockey detects the balance (null) point on the wire.

Wheatstone Bridge Balance Condition

At the null point, no current flows through the galvanometer. The bridge is balanced when:

P / Q = (X + α) / (Y + β)

where α and β are the end corrections at the two ends of the bridge wire, and l is the null-point position measured from the left end.

Deriving the Carey Foster Formula

Let the null point be at distance l1 from the left end when resistance X is in the left gap and Y in the right gap. After interchanging X and Y between the two gaps, let the new null point be at l2.

Applying the Wheatstone Bridge balance condition for both configurations and subtracting, the end corrections α and β cancel out, giving:

Carey Foster Bridge Formula

X − Y = (l1 − l2) × ρ

where:

  • X = unknown resistance (copper wire)
  • Y = known resistance from the resistance box (Ω)
  • l1 = null-point position before interchange (cm)
  • l2 = null-point position after interchange (cm)
  • ρ = resistance per unit length of the bridge wire (Ω/cm)

Determining ρ (Resistance per Unit Length)

Before measuring the unknown resistance, we first calibrate the bridge wire. A standard known resistance R is placed in one gap and a thick copper strip (effectively zero resistance) in the other. The null points l1 and l2 are recorded before and after interchanging. Then:

Formula for ρ

ρ = Rknown / (l1 − l2)

This gives the resistance per centimetre of the bridge wire in Ω/cm.

Connection to Kirchhoff’s Laws

The Carey Foster Bridge operates on the same fundamental principles as the Wheatstone Bridge, which is derived from Kirchhoff’s laws:

  • Kirchhoff’s Current Law (KCL): At each junction (node) of the bridge, the algebraic sum of currents is zero. This ensures current conservation at every node of the circuit.
  • Kirchhoff’s Voltage Law (KVL): Around any closed loop in the bridge network, the algebraic sum of potential differences equals zero. Applying KVL to the two loops of the balanced bridge yields the balance condition P/Q = R/S.

At the null point, the galvanometer current is zero, meaning the potential difference across it is zero. This condition, derived from KVL, is what allows us to compute the unknown resistance from the known values and the null-point readings.

Circuit Diagram

0 50 100 Bridge Wire (100 cm) X (Copper wire) Y (R-box) A B C D P Q E − + K Battery (E) G Jockey l (balance length)

Figure: Carey Foster Bridge circuit diagram showing the bridge wire, unknown resistance X (copper wire), known resistance Y (resistance box), ratio arms P and Q, galvanometer G, battery E with key K, and jockey at the null point.

Procedure

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

  1. Connect the Carey Foster Bridge circuit as shown in the diagram. Place thick copper strips (of negligible resistance) in both gaps X and Y. Ensure all connections are clean and tight.
  2. Close the key K and slide the jockey gently along the bridge wire to locate the null point (where the galvanometer shows zero deflection). Note this reading as l0. It should be close to 50 cm if both strips are identical. This is a preliminary check.
  3. Now place a known standard resistance R (e.g., 0.5 Ω) from the resistance box in gap X, keeping the copper strip in gap Y. Find the null point l1.
  4. Interchange the positions: move the resistance R to gap Y and the copper strip to gap X. Find the new null point l2.
  5. Calculate ρ using the formula: ρ = R / (l1 − l2).
  6. Repeat steps 3–5 for at least three different values of R (e.g., 0.5, 1.0, and 1.5 Ω) and take the mean value of ρ.

Part B: Measurement of Unknown Resistance X (Copper Wire)

  1. Remove the copper strip from gap X and connect the copper wire specimen whose resistance is to be determined.
  2. Place a suitable known resistance Y from the resistance box in gap Y (choose Y close to the expected value of X, typically 0.2–1 Ω for a copper wire).
  3. Close the key and carefully slide the jockey along the bridge wire to locate the null point l1. Record this reading.
  4. Interchange the copper wire and the resistance box between the two gaps. Find the new null point l2. Record this reading.
  5. Calculate the unknown resistance using: X = Y + (l1 − l2) × ρ.
  6. Repeat steps 2–5 with different values of Y (e.g., 0.3, 0.5, 0.7, 0.8, and 1.0 Ω) and calculate the mean value of X.
Important Precautions
  • Always press the jockey gently on the wire — dragging or pressing hard will scratch the wire and cause non-uniformity, leading to errors.
  • Ensure all connections are tight and free of corrosion. Loose contacts introduce stray resistance that distorts the null point.
  • The null point should ideally lie between 30 cm and 70 cm on the bridge wire. If it falls near the ends, adjust the value of Y in the resistance box to bring it closer to the centre for maximum sensitivity.
  • Clean the bridge wire with fine sandpaper before starting the experiment to remove any oxide layer that could affect its uniformity.
  • Switch off the battery (open the key) between readings to prevent heating of the bridge wire, which would change its resistance and introduce systematic error.

Observation Table

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

S.No. Known R (Ω) l1 (cm) l2 (cm) l1 − l2 (cm) ρ = R / (l1 − l2) (Ω/cm)
1 0.5 61.4 38.9 22.5 0.0222
2 1.0 72.1 27.6 44.5 0.0225
3 1.5 77.8 21.5 56.3 0.0266

Mean ρ = (0.0222 + 0.0225 + 0.0266) / 3 = 0.0224 Ω/cm (approx.)

Table B: Measurement of Unknown Resistance X (Copper Wire)

Using ρ = 0.0224 Ω/cm

S.No. Y from R-box (Ω) l1 (cm) l2 (cm) l1 − l2 (cm) X = Y + (l1 − l2) × ρ (Ω)
1 0.3 65.2 49.9 15.3 0.643
2 0.5 56.4 43.0 13.4 0.800
3 0.5 56.8 50.5 6.3 0.641
4 0.7 47.2 50.0 −2.8 0.637
5 0.5 56.5 50.2 6.3 0.641

Mean X = (0.643 + 0.800 + 0.641 + 0.637 + 0.641) / 5

Discarding the outlier (reading 2, where Y was not close enough to X), Mean X ≈ 0.642 Ω

Calculations

Sample Calculation (Reading 1 from Table B)

Given:

  • Y = 0.3 Ω (from resistance box)
  • l1 = 65.2 cm (null point before interchange)
  • l2 = 49.9 cm (null point after interchange)
  • ρ = 0.0224 Ω/cm (determined in Part A)

Using the Carey Foster formula:

X = Y + (l1 − l2) × ρ

X = 0.3 + (65.2 − 49.9) × 0.0224

X = 0.3 + 15.3 × 0.0224

X = 0.3 + 0.3427

X = 0.643 Ω

Taking the mean of all consistent readings (excluding the outlier):

Mean X = (0.643 + 0.641 + 0.637 + 0.641) / 4 = 0.6405 ≈ 0.64 Ω

Result

The electrical resistance of the given copper wire specimen:

X = 0.642 ± 0.04 Ω

The resistance per unit length of the bridge wire:

ρ = 0.0224 Ω/cm

Sources of Error

  • Non-uniform bridge wire: If the bridge wire is not perfectly uniform in cross-section along its length, the resistance per unit length ρ will vary, leading to inaccurate null-point readings and systematic error in all calculations.
  • Loose or corroded connections: High contact resistance at terminals, plugs, or junctions introduces unknown additional resistance into the circuit, shifting the null point and distorting the measured value of X.
  • Heating effect: Prolonged passage of current through the bridge wire and the copper specimen raises their temperature, increasing their resistance. This is especially significant for copper, whose temperature coefficient of resistance is relatively high.
  • Parallax error in reading the scale: If the eye is not positioned directly above the jockey contact point when reading the metre scale, the recorded null-point position will be inaccurate due to parallax.
  • End resistances (stray resistances): Although the interchange method largely eliminates end corrections, any asymmetry in contact resistance between the two configurations can introduce a small residual error in the calculated value of X.

Viva Voce Questions

A Carey Foster Bridge is a modified form of the Wheatstone Bridge designed to measure very small resistances with high precision. It consists of four resistances in a bridge arrangement — two ratio arms P and Q, and two gaps where the unknown and known resistances are placed — connected by a uniform slide wire of known length. The key innovation is the interchange method: by swapping the unknown and known resistances between the two gaps and taking null-point readings in both configurations, the end corrections (stray contact resistances) are eliminated from the final calculation, yielding a much more accurate result than a simple metre bridge.

Interchanging the resistances between the two gaps is the defining technique of the Carey Foster method. Each gap has an unknown end resistance (contact resistance) denoted α and β. When the balance condition is written for both configurations and the two equations are subtracted, the terms containing α and β cancel out completely. This means the final formula X − Y = (l1 − l2) × ρ is independent of these stray resistances, making the measurement self-correcting and far more reliable than a single-configuration measurement.

The main advantage is the elimination of end-correction errors through the interchange method, which a simple metre bridge cannot achieve. In a simple metre bridge, the unknown end resistances at the wire’s contact points directly affect the balance length and cannot be separated from the measured value. The Carey Foster Bridge also offers significantly higher sensitivity when measuring very small resistances (of the order of 0.01 to 1 Ω), where a simple metre bridge would give null points very close to one end of the wire, resulting in poor accuracy and large percentage errors.

The sensitivity of the galvanometer — that is, the rate of change of galvanometer deflection with respect to the jockey position — is maximum near the centre of the bridge wire and drops sharply near the ends. If the null point falls very close to 0 cm or 100 cm, a small movement of the jockey causes a negligible change in the galvanometer deflection, making it extremely difficult to locate the exact balance point. Keeping the null point between 30 and 70 cm ensures that the bridge operates in its most sensitive region, giving the sharpest null and the most precise reading.

Kirchhoff’s laws form the theoretical foundation of all bridge circuits, including the Carey Foster Bridge. Kirchhoff’s Current Law (KCL) states that the total current entering any junction equals the total current leaving it, which ensures current conservation at every node of the bridge network. Kirchhoff’s Voltage Law (KVL) states that the sum of potential differences around any closed loop is zero. Applying KVL to the two loops of the balanced Wheatstone Bridge yields the condition P/Q = R/S, and it is this condition — applied to the Carey Foster configuration with the slide wire — that leads to the working formula used in this experiment.

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