Magnetic Fields & Tangent Galvanometer lab report

1. Title and aim

Magnetic Fields: Tangent Galvanometer, Biot-Savart Law, and Earth's Horizontal Magnetic Component. To investigate the magnetic field generated by an electric current in a circular coil, verify the Biot-Savart law and tangent law of magnetism, observe magnetic torque equilibrium on a compass needle, and determine the horizontal component of Earth's magnetic field (B_H) and the reduction factor using linear regression.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Earth horizontal field B_H of 3.5001e-05 T, differing by 0.003% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• Stewart & Gee Tangent Galvanometer: Precision instrument with 50-turn vertical circular copper coil (mean radius R = 10.0 cm), brass leveling tripod with leveling screws, and central compass box with 360° protractor scale and anti-parallax mirror.
• Pohl Current Reversing Commutator: Heavy-duty four-terminal rocker commutator allowing rapid, arc-free reversal of current direction through the galvanometer coil without disturbing circuit connections.
• Regulated DC Power Supply: Low-noise variable DC voltage supply (0 to 12.0 V, 2 A) equipped with digital voltage and current monitoring.
• Precision Decade Resistance Box / Slide-Wire Rheostat: Non-inductive wire-wound rheostat (0 to 100 Ω, 2 A rating) for smooth, continuous adjustment of coil current.
• Digital Multimeter / Precision Ammeter: Calibrated 4½-digit true-RMS digital ammeter (0.1 mA resolution) to verify current readings against galvanometer deflections.
• Magnetic Dip Needle and Compass: Freely suspended dip circle for determining the local magnetic meridian and measuring inclination angle.
• Ambient room temperature: T = 20.0 °C; Reference Earth horizontal field: B_H = 3.500 × 10⁻⁵ T (35.00 μT).
• Galvanometer coil: N = 50 turns of insulated copper wire (mean radius R = 10.00 cm).
• Coil magnetic constant: G = μ₀ N / (2R) = 3.14159 × 10⁻⁴ T A⁻¹.
• Galvanometer reduction factor: K = B_H / G = 0.11141 A (111.4 mA).
• Magnetic needle: High-coercivity cobalt steel needle (length 2ℓ = 15.0 mm ≪ 2R) with jeweled pivot.
• Permeability of free space: μ₀ = 4π × 10⁻⁷ T m A⁻¹ = 1.256637 × 10⁻⁶ T m A⁻¹.

4. Theory and graph choice

The discovery by Hans Christian Ørsted in 1820 that an electric current deflects a magnetic needle inaugurated the unified science of electromagnetism. Soon after, Jean-Baptiste Biot and Félix Savart formulated the fundamental law governing the magnetic field dB produced by an infinitesimal current element I dℓ: dB = (μ₀ / 4π) [I (dℓ × r̂) / r²], where μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space. Integrating this fundamental relation around a flat circular coil of N turns and mean radius R carrying direct current I establishes an axial magnetic field at the coil's geometric center: B_coil = (μ₀ N I) / (2R) = G I, where G = (μ₀ N) / (2R) is the geometric coil constant. In the tangent galvanometer, this circular coil is mounted vertically with its plane aligned strictly in the magnetic meridian—parallel to the horizontal component of Earth's geomagnetic field B_H. At the exact center of the coil, a small magnetic compass needle is pivoted on a low-friction jewel bearing. When direct current passes through the windings, the needle experiences two mutually orthogonal magnetic fields: Earth's horizontal field B_H along the plane of the coil, and the current's magnetic field B_coil perpendicular to the coil plane. Under the action of these orthogonal fields, the needle experiences opposing magnetic torques: τ_coil = M B_coil cos θ and τ_Earth = M B_H sin θ, where M is the magnetic dipole moment of the needle and θ is the angular deflection. Equating these torques establishes the celebrated tangent law of magnetism: B_coil = B_H tan θ. Substituting the Biot-Savart expression yields: (μ₀ N I) / (2R) = B_H tan θ, which rearranges to tan θ = [μ₀ N / (2 R B_H)] I, or I = [2 R B_H / (μ₀ N)] tan θ = K tan θ, where K = B_H / G is the reduction factor of the galvanometer. By systematically varying the current I using a rheostat and measuring the resulting deflection angle θ across four quadrant orientations using an anti-parallax mirror and a reversing commutator, students cancel zero-offset and alignment errors. Ordinary least-squares regression of tan θ against current I yields a straight line whose slope m = G / B_H isolates Earth's horizontal field B_H = G / m with remarkable precision, confirming the foundations of electromagnetic force, geomagnetic field metrology, and Ampère's circulating currents.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units A⁻¹.

5. Method

1. Place the tangent galvanometer on a rigid, non-magnetic wooden bench away from iron fixtures or DC power leads. Using the leveling screws, level the base until the central compass needle swings freely. Rotate the coil until its vertical plane is exactly parallel to the compass needle pointing North-South (0°–0° alignment).
2. Connect the DC power supply in series with the rheostat, precision ammeter, reversing commutator, and the 50-turn coil terminals. Inspect all connections to verify tight, clean non-magnetic contacts.
3. Turn on the power supply and adjust the rheostat until a current of I = 50.0 mA flows through the coil. Using the anti-parallax mirror (aligning the pointer with its image), record the deflection angle θ1 on both ends of the aluminum pointer (East and West tips).
4. Throw the reversing commutator to reverse the current direction through the coil without altering the rheostat. Record the deflection angle θ2 at both ends of the pointer on the opposite side of the scale, and calculate the mean deflection θ = (θ1 + θ2) / 2.
5. Repeat the four-reading measurement sequence for six discrete current levels (50.0, 80.0, 110.0, 140.0, 170.0, and 200.0 mA). Record current I in mA and calculate tan θ for each station.
6. Measure the inner and outer diameters of the circular coil using vernier calipers to compute the mean radius R = 10.0 cm, and confirm the turn count N = 50.

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0030%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 3.7283e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 7.6928e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.0392e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 2.9998e-09 T. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Compass pivot agate jewel frictional resistance (stiction threshold ±0.2° on needle deflection), anti-parallax mirror alignment visual resolution, stray DC magnetic fields from power cords or ferromagnetic building steel reinforcement (minimized by twisting power supply leads and isolating non-magnetic brass bench), coil radius measurement tolerance (inner and outer vernier caliper error ±0.1 mm producing ±0.1% uncertainty in G), winding axial spread (finite coil thickness vs ideal infinitesimal filament), and diurnal solar geomagnetic drift (~0.05 μT during laboratory duration) contribute experimental uncertainty. Bilateral pointer reading (averaging East and West tips) and current commutation across four stations cancel pointer zero-offset bending and residual magnetic meridian alignment errors, while ordinary least-squares regression of tan θ against I provides an unbiased slope isolating Earth horizontal field B_H.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Ensure no ferromagnetic objects (iron stands, steel screwdrivers, cell phones, or transformer cores) are within 1.5 meters of the apparatus to prevent stray magnetic distortion.
• Align the aluminum pointer tip with its reflection in the mirror ring directly beneath it to eliminate visual parallax when reading the degree scale.
• Keep currents below 300 mA to prevent resistive Joule heating of the copper coil, which would alter coil dimensions and wire resistance.
• Always reverse current and average all four pointer readings (both ends, both polarities) for every observation station.

The analytical model treats the magnetic needle as an infinitesimal point dipole situated at the exact geometric center of the circular coil, assumes perfectly uniform coil windings with negligible axial width, and treats Earth's magnetic field as strictly stationary and uniform. In a physical laboratory, finite needle length introduces a small second-order correction (1 + 0.75 (ℓ/R)² (5 cos² θ − 1) ≈ 0.2%), building structural steel reinforcement produces localized geomagnetic distortion, and diurnal solar geomagnetic drift causes minor baseline fluctuations (~0.1 μT).

12. Conclusion

The worked example illustrates how to obtain Earth horizontal field B_H from measurements, graph variables and a fitted coefficient. The estimate is 3.5001e-05 T. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.