Magnetic Moments & Magnetometer lab report

1. Title and aim

Magnetic Moments: Gauss Deflection Magnetometer, Dipole Fields, and Tan-A Position. To investigate the magnetic dipole field generated by permanent bar magnets along the axial End-on (Tan-A) position of Gauss, verify the inverse-cube magnetic field distance dependence (B ∝ 1/d³), determine the magnetic dipole moment (M) and the ratio of moments using ordinary least-squares linear regression, and compare deflection and oscillation methods for geomagnetic metrology.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Magnetic dipole moment M of 0.8750 A m², differing by 0.002% 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

• Gauss Deflection Magnetometer (Tan-A Metrology Bench): Rigid 1-meter non-magnetic mahogany optical bench with dual millimeter scales, leveling tripod screws, and central circular compass box with 360° protractor scale, anti-parallax mirror ring, and jewel-pivoted needle.
• Primary Alnico Bar Magnet (Magnet 1): Precision cylindrical permanent magnet (length 2L = 10.0 cm, diameter 8.0 mm, dipole moment M₁ = 0.8750 A m²) for primary inverse-cube field mapping and slope regression.
• Secondary Comparison Bar Magnet (Magnet 2): Matched cylindrical permanent magnet of lower magnetization (M₂ = 0.5250 A m²) for evaluating magnetic moment ratios (M₁ / M₂) across identical distance stations.
• Reversible Sliding Magnet Carrier: Non-magnetic brass sliding carriage equipped with vernier index cursor, thumbscrew lock, and dual orientation V-blocks for rapid, repeatable magnet positioning and 180° axial reversal.
• Vibration Magnetometer Enclosure: Draft-free glass-walled oscillation chamber with unspun silk fiber suspension and torsion head for measuring the magnet's torsional oscillation period T in Earth's field.
• Digital Stopwatch and Precision Vernier Calipers: Quartz digital timer (0.01 s resolution) for oscillation timing and vernier calipers (0.02 mm resolution) for measuring magnet physical dimensions to compute moment of inertia.
• Ambient room temperature: T = 20.0 °C; Reference Earth horizontal field: B_H = 3.500 × 10⁻⁵ T (35.00 μT).
• Primary test magnet (Magnet 1): High-coercivity Alnico bar magnet (physical length 2L = 10.00 cm, magnetic length 2ℓ = 8.40 cm, mass m = 38.20 g).
• Secondary comparison magnet (Magnet 2): Matched Alnico bar magnet of lower magnetization (M₂ = 0.5250 A m²).
• Magnetometer needle: Ultra-short cobalt steel compass needle (length 2ℓ_n = 10.0 mm ≪ d) on low-friction agate jewel pivot.
• Permeability of free space constant: μ₀ / (4π) = 1.000 × 10⁻⁷ T m A⁻¹.
• Reference regression slope: m_ref = (2 × 10⁻⁷ × 0.8750) / (3.500 × 10⁻⁵) = 5.000 × 10⁻³ m³.

4. Theory and graph choice

The concept of the magnetic dipole moment forms the cornerstone of classical magnetostatics and atomic magnetic phenomena. Any magnetized body, such as a permanent bar magnet or an electric current loop, possesses a magnetic dipole moment vector M. For a bar magnet having magnetic pole strength p and effective magnetic pole separation 2ℓ (where physical length 2L and magnetic length 2ℓ are related empirically by 2ℓ ≈ 0.84 × 2L due to pole internalization), the magnetic moment is defined as M = p (2ℓ), directed internally from the magnetic South pole to the magnetic North pole. In 1832, Carl Friedrich Gauss formulated the classical magnetometer methods for determining magnetic fields and dipole moments in absolute units, establishing two canonical orthogonal geometries: the End-on position (Tangent Position A or Tan-A) and the Broadside-on position (Tangent Position B or Tan-B). In the Tan-A configuration, a small magnetic compass needle is pivoted freely on an agate jewel bearing at the center of a circular graduated protractor. A horizontal metric bench extends symmetrically East-West, strictly perpendicular to the local magnetic meridian. When a bar magnet of moment M is positioned on the bench at distance d from the compass pivot with its axis aligned along the East-West line, the axial magnetic field B produced by the magnet at the needle's location is given by Coulomb's magnetic force law as: B = (μ₀ / 4π) [2 M d / (d² − ℓ²)²], where μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space. When the separation distance d is large compared to the magnet's half-length (d ≫ ℓ), the higher-order multipole term (d² − ℓ²)² can be expanded via the binomial theorem, yielding the leading-order inverse-cube dipole field: B ≈ (μ₀ / 4π) (2M / d³). At the magnetometer center, the pivoted needle experiences two mutually orthogonal horizontal magnetic fields: Earth's horizontal geomagnetic field B_H along the North-South magnetic meridian, and the magnet's axial field B along the East-West line. By the tangent law of magnetism, the needle aligns along the resultant vector at an angle θ such that B = B_H tan θ. Equating the field expressions yields: (μ₀ / 4π) (2M / d³) = B_H tan θ, which rearranges directly to: tan θ = [(μ₀ / 4π) (2M / B_H)] (1 / d³). By designating x = 1 / d³ (in m⁻³) as the independent variable and y = tan θ as the dependent variable, ordinary least-squares regression fits the canonical straight line y = m x + b, where the slope m = (μ₀ / 4π) (2M / B_H) isolates the magnetic dipole moment M = (m B_H) / [2 (μ₀ / 4π)] with exceptional accuracy. By systematically reversing the magnet's polarity and testing both East and West arms, students eliminate zero-offset pointer bending, pivot eccentricity, and residual meridian misalignment, confirming the fundamental foundations of dipolar magnetostatics and completing the classical study of electric and magnetic fields.

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 m³.

5. Method

1. Place the deflection magnetometer on a rigid non-magnetic bench far from ferromagnetic fixtures or DC power leads. Using the brass leveling screws, level the instrument until the compass needle swings freely. Rotate the entire bench until the long aluminum pointer reads exactly 0°–0° while the arms of the bench extend precisely East and West.
2. Measure the physical length 2L and diameter of Magnet 1 using vernier calipers, and weigh it on an analytical balance. Calculate the effective magnetic length between pole centers using the standard empirical relation 2ℓ = 0.84 × 2L = 8.40 cm.
3. Place Magnet 1 on the East arm carrier at distance d = 14.0 cm with its North pole facing inward toward the compass needle. Read both ends of the aluminum pointer (θ₁ and θ₂) using the anti-parallax mirror. Reverse the magnet on the carrier so its South pole faces inward, and record both ends (θ₃ and θ₄).
4. Transfer the magnet carrier to the West arm at the exact same distance d = 14.0 cm. Repeat the dual-polarity measurement sequence (North inward, then South inward), recording both pointer ends (θ₅, θ₆, θ₇, θ₈). Compute the grand mean deflection θ = (Σ θ_i) / 8.
5. Repeat the complete eight-reading measurement sequence for six discrete distances (d = 14.0, 16.0, 18.0, 20.0, 22.0, and 24.0 cm). For each distance, compute the inverse-cube variable x = 1 / d³ (in m⁻³) and the dependent variable y = tan θ.
6. Replace Magnet 1 with Magnet 2 and record deflections at identical distances. Compute the ratio of magnetic moments M₁ / M₂ = tan θ₁ / tan θ₂ to verify relative magnetization independent of Earth's field B_H.

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 = 0.0022%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 8.7940e-08; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 6.0424e-07. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.2371e-04.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1.0574e-04 A m². This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Magnetometer pivot jewel stiction threshold (±0.2° dead-band resolution in needle response), visual parallax against the anti-parallax mirror ring, finite bar magnet length corrections (the exact axial field B = (μ₀/4π)[2Md/(d²−ℓ²)²] differs from the ideal dipole 2M/d³ by factor [1 − (ℓ/d)²]⁻² ≈ +1.4% at closest distance d = 14 cm with 2ℓ = 8.4 cm), stray environmental magnetic fields from building structural steel or DC wiring, magnet sliding carrier indexing tolerance (±0.5 mm in distance d, magnified cubically as 3(δd/d) ≈ 1.1% in 1/d³), and diurnal geomagnetic baseline drift (~0.05 μT) govern the uncertainty envelope. Collecting eight readings across four reversal quadrants (East/West arms and direct/reversed polarity) cancels pointer bending, pivot eccentricity, and residual meridian alignment offsets, while ordinary least-squares regression of tan θ against 1/d³ yields an unbiased slope isolating the true magnetic dipole moment M.

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 materials (steel rulers, iron stands, screwdrivers, cell phones, or wristwatch steel bands) are within 1.5 meters of the magnetometer bench to prevent magnetic distortion.
• Always align the aluminum pointer tip with its optical reflection in the mirror ring directly beneath it to eliminate visual parallax when reading the degree scale.
• Use the complete eight-reading reversal sequence (both ends, both polarities, both arms) for every distance station to eliminate pivot eccentricity and pointer zero-offset errors.
• Handle the permanent bar magnets gently; never drop, strike, or heat them, as mechanical shock and elevated temperatures cause domain wall relaxation and irreversible demagnetization.

The theoretical model treats the magnetic needle as an infinitesimal point dipole situated at the exact geometric center of the bench, assumes the bar magnet acts as an ideal point dipole (neglecting finite magnetic length corrections (d² − ℓ²)² which contribute ~0.5% at closest approach d = 14 cm), and treats Earth's magnetic field as strictly stationary and uniform. In a physical laboratory, building structural steel reinforcement produces localized geomagnetic distortion, pivot jewel stiction introduces a ±0.2° dead-zone threshold, and diurnal geomagnetic drift causes minor baseline fluctuations (~0.1 μT).

12. Conclusion

The worked example illustrates how to obtain Magnetic dipole moment M from measurements, graph variables and a fitted coefficient. The estimate is 0.8750 A m². A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.