Definition
A magnetometer is an instrument that measures a magnetic field. In most physics contexts, the measured quantity is the magnetic flux density \(B\), expressed in tesla (T), often resolved into components (for example, a horizontal component near Earth’s surface).
Units used in magnetometry: \(1\ \text{T} = 1\ \text{N}\cdot\text{s}/(\text{C}\cdot\text{m})\) and \(1\ \text{G} = 10^{-4}\ \text{T}\). Many natural fields (Earth’s field) are conveniently reported in \(\mu\text{T}\).
Classical-mechanics viewpoint: a magnetometer as a torque-and-motion problem
A mechanically intuitive magnetometer model treats a small magnet (or equivalent sensor element) as a magnetic dipole with dipole moment \(\boldsymbol{\mu}\) mounted so it can rotate. The field \(\boldsymbol{B}\) exerts a torque, producing angular motion governed by rotational dynamics.
Core formulas
1) Magnetic potential energy
\[ U(\theta) = -\mu \cdot B \cdot \cos(\theta) \]
Here \(\theta\) is the angle between \(\boldsymbol{\mu}\) and \(\boldsymbol{B}\). The lowest energy occurs at \(\theta = 0\), when the dipole aligns with the field.
2) Torque on a magnetic dipole (vector and magnitude)
\[ \boldsymbol{\tau} = \boldsymbol{\mu}\times \boldsymbol{B} \qquad \Rightarrow \qquad \tau = \mu \cdot B \cdot \sin(\theta) \]
3) Rotational equation of motion
For a rotating body with moment of inertia \(I\) about the pivot axis, \[ I \cdot \ddot{\theta} = -\mu \cdot B \cdot \sin(\theta) \]
The negative sign indicates a restoring torque that tends to reduce \(\theta\) (bring the dipole back toward alignment).
Small-angle approximation and a practical magnetometer formula
For small deflections, \(\sin(\theta)\approx \theta\) (with \(\theta\) in radians). The equation becomes a simple harmonic oscillator:
\[ I \cdot \ddot{\theta} = -\mu \cdot B \cdot \theta \]
Comparing with \(\ddot{\theta} = -\omega^2 \cdot \theta\) gives \[ \omega = \sqrt{\frac{\mu \cdot B}{I}} \qquad \Rightarrow \qquad T = \frac{2\pi}{\omega} = 2\pi \cdot \sqrt{\frac{I}{\mu \cdot B}} \]
Solving for the magnetic field produces a direct working formula (used in vibrating/oscillation-based magnetometer ideas):
\[ B = \frac{4\pi^2 \cdot I}{\mu \cdot T^2} \]
Worked example (oscillation method)
A small bar magnet is suspended so it can oscillate in a uniform field component \(B\). Suppose: \(I = 1.00\times 10^{-4}\ \text{kg}\cdot\text{m}^2\), \(\mu = 0.500\ \text{A}\cdot\text{m}^2\), and the measured period is \(T = 12.6\ \text{s}\). Then
\[ B = \frac{4\pi^2 \cdot (1.00\times 10^{-4})}{0.500 \cdot (12.6)^2} = \frac{4\pi^2 \cdot 1.00\times 10^{-4}}{0.500 \cdot 158.76} \approx 4.97\times 10^{-5}\ \text{T} \approx 49.7\ \mu\text{T} \]
A value near \(50\ \mu\text{T}\) is consistent with the order of magnitude of Earth’s magnetic field.
Quick reference table
| Quantity | Symbol | Formula | Typical unit |
|---|---|---|---|
| Magnetic flux density | \(B\) | Measured by a magnetometer | \(\text{T}\), \(\mu\text{T}\) |
| Dipole potential energy | \(U\) | \(U(\theta) = -\mu \cdot B \cdot \cos(\theta)\) | \(\text{J}\) |
| Dipole torque magnitude | \(\tau\) | \(\tau = \mu \cdot B \cdot \sin(\theta)\) | \(\text{N}\cdot\text{m}\) |
| Oscillation period (small angle) | \(T\) | \(T = 2\pi \cdot \sqrt{\frac{I}{\mu \cdot B}}\) | \(\text{s}\) |
| Field from period (small angle) | \(B\) | \(B = \frac{4\pi^2 \cdot I}{\mu \cdot T^2}\) | \(\text{T}\) |
Visualization: dipole torque and small oscillations
Common pitfalls
- Confusing \(B\) with magnetic field strength \(H\): many instruments report \(B\) (tesla), while some contexts use \(H\) (A/m); the relation depends on material response.
- Using degrees in the small-angle approximation: \(\sin(\theta)\approx \theta\) requires \(\theta\) in radians.
- Ignoring which component is measured: a rotating dipole often responds to the component of \(\boldsymbol{B}\) that provides a restoring torque about the pivot axis (for Earth-field setups, typically a horizontal component).
Final takeaway
The keyword magnetometer refers to a magnetic-field measuring instrument; in a classical-mechanics model the essential link is the dipole torque \[ \boldsymbol{\tau} = \boldsymbol{\mu}\times \boldsymbol{B} \] and, for small oscillations, \[ B = \frac{4\pi^2 \cdot I}{\mu \cdot T^2}. \]