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The PHYWE guide Magnetic moment in the magnetic field makes the magnetic dipole moment measurable. A circular test current loop sits between a pair of Helmholtz coils, with separate supplies and meters for the test electric current $I$ and field-producing electric current $I_H$. The test current loop is mounted on a torsion dynamometer. The guide varies $I_H$, the number of turns, the test electric current, diameter, and the angle between the magnetic dipole moment and magnetic field. Its angular settings are spaced by $15°$. The leads hang loosely and are twisted together to suppress extra torque; the instrument zero is checked frequently because moving leads can shift it. Its reported results include approximately linear dependence on each electric current and on winding count, and approximately quadratic dependence on diameter.^[ref]

The readout is a balance of torques. According to the manufacturer's Torsion dynamometer 0.01 N operating instructions, a metallic band supplies the restoring torque. First set the indication to zero with the applied force absent and center the lever using the zero knob. Apply the load, then turn the indicating knob until the lever returns to its original zero position. The scale then reads a force equivalent, not a torque directly. The quoted scale division is $0.1\mathrm{mN}$. For a force applied perpendicular to the calibration lever arm at distance $\ell$, the conversion is $\tau =F\ell$; a current loop applies a couple, so its torque must be compared with the calibrated restoring torque.^[ref]

For a quantitative repeat, independently check that conversion with known forces at a measured perpendicular lever arm, and check the imposed magnetic field at the test current loop. These are proposed calibration checks, not additional measurements reported here. The ideal Helmholtz coil model predictswhere $N_H$ and $R_H$ are the winding count and radius of each field-producing Helmholtz coil, distinct from $N$ and $R$ of the test current loop. Derive $B_0$ by adding the on-axis magnetic field of a circular current loop at distances $R_H/2$ from both coils. Independently calibrating $B_0$ and the restoring torque lets a fitted torque amplitude determine $m$ as amplitude divided by $B_0$, to compare with $NI\pi R^2$.

Useful additional controls are to reverse only $I$, reverse only $I_H$, and reverse both: the signed torque should reverse in either single reversal and be unchanged in the double reversal. Record baselines with each electric current separately zero, and subtract a nearby zero reading to reduce torsion balance zero drift. Repeat angular settings in both directions to reveal lead-dependent offsets. These are suggested controls. Limits include finite magnetic field uniformity, uncertainty in effective winding area and angular zero, meter calibration, Joule heating, and residual lead torque. An apparently sinusoidal trace alone does not establish the absolute magnetic dipole moment. No raw laboratory data or new measured values are claimed here.

The University of Iowa's Electron Spin Resonance, University of Iowa laboratory protocol (revised July 2007, Experiment and Resonance Data Analysis, printed pp. 2–6) specifies DPPH on cotton in a tube inside an RF coil, between Helmholtz coils. A bridge monitors RF absorption; an oscilloscope records this output and the voltage across a $1\Omega$ current-monitor resistor. Sweeping the coil current passes through the resonant magnetic field. The protocol describes two dips when the sweep revisits resonance; these are repeated crossings, not two distinct spectral lines.

Its calibration replaces the sample with a Hall effect probe, maps the magnetic field in $2\mathrm{mm}$ steps, and accounts for sample dimensions. At least four RF frequencies test the resonance relation. Sweep speed is adjusted to obtain comparable passages. Only a local peak-plus-background fit is prescribed, because the full shape is not Gaussian. The protocol also flags unexplained trailing spikes and possible electronics contributions. These controls constrain field and sweep errors; an empty-tube/cotton blank would be an additional background control, not a reported measurement in this protocol.^[ref]

For an isolated effective spin one-half with dimensionless positive factor $g$, the basic resonance model isHere ${\mu }_{\mathrm{B}}$ is the Bohr magneton, $h$ is the Planck constant, $\nu$ is the applied frequency, and $B_{\mathrm{res}}$ is the calibrated resonant magnetic field. A change in frequency should shift the resonance field proportionally in this model. The factor $g$ describes the electronic magnetic moment coupling, not the species' net electric charge.

A resonance supports the presence of detectable paramagnetism. It does not, by itself, identify an unknown radical: other species exhibiting paramagnetism, impurities, overlapping responses, and interactions with atomic nuclei can complicate interpretation. Conversely, absence of a detectable resonance need not mean absence of free radicals; a low concentration, a short lifetime, or an unsuitable measurement range can conceal them. The occupancy and chain diagrams are original bookkeeping illustrations; neither is a measured or simulated DPPH spectrum.

A Helmholtz coil arrangement uses two identical coaxial circular windings separated by their radius, carrying equal electric currents in the same circulation sense. Their combined magnetic field is especially uniform near the midpoint, making it useful for a current-loop torque experiment. The cancellation of the axial second derivative there follows from the on-axis magnetic field of a circular current loop.

Alfred Kastler's 1966 lecture recalls his 1950 proposal and distinguishes the early ${}^{23}\mathrm{Na}$ atomic beam work from later sealed-vapour experiments. Its Fig. 5 shows subsequent sodium radio wave resonances, including multiple-photon transitions.^[ref]

The 1952 demonstration by Jean Brossel, Alfred Kastler and Jacques Winter used a sodium atomic beam, lamp light with circular polarization and fluorescence readout. Jean Brossel was a central experimental contributor; Alfred Kastler received the 1966 Nobel Prize in Physics alone.^[ref][ref]

The lecture also describes a ${}^{199}\mathrm{Hg}$ experiment: a pumping beam with circular polarization prepares the vapour, while a perpendicular probe converts transverse magnetic dipole moment Larmor precession into modulated absorption at a photodetector. Interrupting pump and RF together exposes decaying free Larmor precession. Its envelope measures transverse relaxation under that model; it is distinct from the sodium atomic beam fluorescence signal.^[ref]

MIT's later rubidium experiment pumps natural ${}^{85}\mathrm{Rb}$/${}^{87}\mathrm{Rb}$ with a discharge lamp filtered to D1 ($5S_{1/2}\to 5P_{1/2}$, near $795\mathrm{nm}$). A polarizer and quarter-wave plate precede a heated rubidium/neon cell. Helmholtz coils set a weak static magnetic field. Separate transverse RF mixing increases absorption, measured by photodiode transmission.^[ref]

Compare resonant and detuned RF with the optical beam fixed, then change the calibrated static magnetic field: a true magnetic sublevel resonance should move with its Zeeman splitting. Lamp drift, stray magnetic fields, imperfect circular polarization, optical attenuation and RF power broadening can change the observed signal. Consequently, a transmission dip alone does not determine every sublevel population. These are measurement-design considerations, not additional reported historical observations.

Put the two coil centres at $z=\pm a/2$. By superposition, their axial magnetic field isBoth magnetic field contributions point the same way. Evaluation givesThe corresponding coefficients are $0.715542$ and $0.676777$. Each individual magnetic field falls as one moves away from its own coil; their sum is much flatter near the midpoint.

For a single coil, differentiating $(a^2+x^2{)}^{-3/2}$ twice produces $3(4x^2-a^2)(a^2+x^2{)}^{-7/2}$. At $x=a/2$ this vanishes. Thus the pair has $B^{\prime }(0)=0$ by reflection symmetry and $B^{\prime \prime }(0)=0$ by its chosen separation; all odd derivatives vanish there. In fact,This Helmholtz coil arrangement produces a nearly uniform central magnetic field, useful for controlled charged particle in a uniform magnetic field and magnetic field calibration. The small orbit used below lies within that central region.

For a steady thin-wire electric current, the Biot-Savart law iswhere $\mathbf{R}$ points from the source element to the field point and $d\mathbf{\ell }$ follows positive electric current. On a circular loop's axis the transverse contributions cancel by symmetry. The axial numerator is $a^{2}d\phi$ and the distance is $\sqrt{a^2+z^2}$, givingIts graph is positive and even, with maximum ${\mu }_{0}I/(2a)$ at the centre and a $|z{|}^{-3}$ far-field decay. Reversing the electric current reverses its sign.

For two equally oriented $N$-turn coils centred at $\pm D$, superposition givesConsequentlyThis is an even differentiable function, so every odd derivative at zero vanishes, including the first and third. To check the second derivative explicitly, let $f(s)=(a^2+s^2{)}^{-3/2}$. ThenThus $2D=a$ makes the quadratic variation vanish, which is the Helmholtz coil condition. The central value and leading axial variation are

Near the centre, the coils' magnetic field lines are nearly straight and parallel to the axis. On the axis rotational symmetry makes $B_x=B_y=0$ exactly; in the central plane the radial components from the two coils cancel. More quantitatively, in the current-free central volume, axisymmetry and $\nabla \cdot \mathbf{B}=0$ implyThe suppressed axial derivatives therefore also suppress transverse magnetic fields near the centre. The smallness applies locally compared with the coil radius; it is not a claim of an exactly uniform field throughout the apparatus.