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A moving-coil galvanometer has a winding that rotates about an axis, a restoring spring, and a pointer or optical readout. In the radial-field construction, a cylindrical iron core and shaped magnetic poles keep the magnetic field approximately in the coil's plane as it turns. The resulting moving-coil galvanometer torque is nearly independent of the deflection angle; the spring then converts electric current into a proportional settled deflection. ^[ref]

For a rectangular winding with $N$ turns, active side length $\ell$, side separation $w$, and per-turn area $A=w\ell$, the magnetic force on a current-carrying wire gives equal and opposite side forces of magnitude $N|I|\ell B$. Their force couple has lever arm $w$. Choosing the positive rotation to match positive electric current in the illustration gives the signed torque and spring balanceHere $B$ is the working-gap magnetic flux density, $k>0$ is the torsional spring constant in $\mathrm{N}m/rad$, and $\theta$ is the deflection angle in radians from the no-current zero. The balance holds after motion has settled, within the radial-field and linear elasticity ranges. Do not identify $\theta$ with the angle between a magnetic dipole moment and a fixed uniform magnetic field: that different geometry has torque magnitude $NIAB|\sin \alpha |$. ^[ref]

For an original illustrative design, choose $N=120$, $A=3.00\times 10^{-4}{\mathrm{m}}^2$, $B=0.250\mathrm{T}$, and $k=3.00\times 10^{-6}\mathrm{N}m/rad$. Then $NAB=9.00\times 10^{-3}\mathrm{N}m/A$. At $I=+40.0\mu \mathrm{A}$, the magnetic torque is $+3.60\times 10^{-7}\mathrm{N}\mathrm{m}$ and $\theta =+0.120\mathrm{rad}\simeq +6.88^{\circ }$. Reversing the electric current reverses both the torque and the deflection, giving $-6.88^{\circ }$ without changing the magnetic field.

A physicist.^[ref]

For a smooth axisymmetric magnetic field, write $B_z(\rho ,z)=B_0(z)+O({\rho }^2)$. The absence of magnetic monopoles gives ${\rho }^{-1}{\partial }_{\rho }(\rho B_{\rho })=-B_0^{\prime }(z)+O({\rho }^2)$. Integrating from the axis and excluding a singular $1/\rho$ term proves the displayed relation. An axial field decreasing along $+z$ has an outward radial component, even when the field points purely axially on the axis itself.

The Néel temperature marks loss of long-range antiferromagnetism. A Curie temperature concerns ferromagnetism or ferrimagnetism; magnetic compensation temperature instead denotes cancellation within an ordered ferrimagnet.^[ref]

In the simple collinear description, compare $|\mathbf{L}|$ across $T_N$, rather than expecting $|\mathbf{M}|$ to switch from nonzero to zero. Zero-field magnetization may be zero on both sides. A nonzero magnetic susceptibility is also compatible with antiferromagnetism: an applied magnetic field can disturb the cancellation. The uniform response of a two-sublattice Ising antiferromagnet provides a model calculation.

For charges $+e,-e$ and relative coordinate $\mathbf{r}={\mathbf{r}}_2-{\mathbf{r}}_1$, total magnetic pseudomomentum is conserved in a uniform magnetic field. In an appropriate symmetric-gauge Lagrangian differing by a total derivative, it is the canonical centre-of-mass momentum. Mechanical centre-of-mass momentum instead equals $\mathbf{K}+e\mathbf{B}\times \mathbf{r}$ and need not be constant. The sector $\mathbf{K}=0$ is not generally a stationary-centre-of-mass frame.

For an ideal infinite filament with zero line charge density and current $I$ in a laboratory frame, the line-integrated four-current is $(0,I)$ in temporal/axial components. An axial boost of speed $u$ gives the title density and current. The laboratory magnetic field ${\mu }_{0}I\hat{\mathbf{\phi }}/(2\pi r)$ therefore becomes both an electric field from ${\lambda }^{\prime }$ and a magnetic field from $I^{\prime }$. Charge neutrality is frame-dependent for a current-carrying system. A model built from oppositely charged moving and resting populations makes this result explicit; the charge of each fixed material particle set remains invariant.

A neutron star is a compact stellar remnant whose gravity compresses roughly a stellar mass into a radius of order ten kilometres. Degenerate matter supplies most of its pressure support, while rapid rotation and intense magnetic fields can power observable emission.

An electrically neutral neutron still has a magnetic moment $\mathbf{\mu }={\gamma }_n\mathbf{S}$. A collimated beam passing through an inhomogeneous magnetic field experiences the force derived from $H=-\mathbf{\mu }\cdot \mathbf{B}$. In a region whose field axis is approximately fixed, a spin one-half neutron has two field-axis spin eigenstates, producing two allowed transverse forces and two separated beam components after a drift. A homogeneous field can cause Larmor precession but cannot separate positions through this force.

The neutron's negative gyromagnetic ratio interchanges which spin projection moves toward increasing field; it does not change the number of branches. An unpolarized beam gives equal ideal populations. A beam prepared in one field-axis eigenstate populates only one branch, so splitting into two observed spots presupposes both components and sufficient spatial resolution.

A finite-amplitude ideal-MHD Alfvén wave can rotate the transverse magnetic field at constant magnitude, with constant mass density, pressure and longitudinal components. Its transverse velocity obeys ${\mathbf{u}}_{\perp }=\mathbf{C}-s{\mathbf{B}}_{\perp }/\sqrt{{\mu }_0\rho }$ and its speed is constant along the wave. The constant magnetic pressure prevents compressive acceleration.

A cataclysmic variable whose white dwarf magnetic field does not dominate the large-scale accretion flow. It normally supports an accretion disk fed through Roche-lobe overflow. The term does not rule out a magnetized donor star or magnetic braking of a binary star.

A normal-field wall imposes zero tangential magnetic field: $\mathbf{B}\times \mathbf{n}=0$ for the wall unit normal $\mathbf{n}$. In a channel with imposed normal field $B_0\hat{\mathbf{z}}$, an induced tangential component $b(z)\hat{\mathbf{x}}$ therefore satisfies $b=0$ on each wall. This condition is distinct from the no-slip boundary condition on fluid velocity.

The magnetic dipole moment of an atomic nucleus. Its relation to nuclear spin is expressed by the gyromagnetic ratio; its interaction with an applied magnetic field underlies nuclear magnetic resonance.

Nuclear magnetic resonance detects the response of atomic nuclei with a magnetic moment to a radio frequency magnetic field in a steady magnetic field. Its striking possibility is to measure a nuclear property through an ordinary electrical signal from a bulk sample: an enormous number of tiny magnetic moments can produce a measurable change in a receiving circuit.

For a nucleus with spin angular momentum $\mathbf{J}$, define its gyromagnetic ratio by $\mathbf{\mu }=\gamma \mathbf{J}$. Throughout this article $\gamma$ is angular, measured in radians per second per tesla; $\gamma /(2\pi )$ is the corresponding value in cycles per second per tesla. With ${\mathbf{B}}_0=B_0\hat{\mathbf{z}}$, the Hamiltonian is $H=-\gamma B_0{J}_z$. Adjacent energy levels differ byThus a measured resonance frequency and a calibrated magnetic field determine $|\gamma |$. Converting that into a nuclear magnetic moment also requires the spin quantum number: the maximum projection has magnitude $|\gamma |\hslash I$. This simple experiment does not by itself determine the sign of $\gamma$.

The Rabi molecular beam magnetic resonance experiment detects transitions through changes in a selected molecular beam. The bulk experiments below instead detect electrical consequences of nuclear magnetization in matter at ordinary density. A small Boltzmann distribution population imbalance is multiplied by a very large number of nuclei. For noninteracting spin one-half nuclei of number density $n$ and positive $\gamma$, their equilibrium magnetization isThis follows by weighting the two projections $\pm \gamma \hslash /2$ with their Boltzmann distribution probabilities. It explains both the smallness of the bulk polarization at ordinary temperature and the value of many nuclei contributing to one receiving circuit.

NMR spectroscopy measures resonant responses of atomic nuclei to applied electromagnetic radiation in a magnetic field. In residual-group monitoring of polyurethane formation, distinguishable signal integrals supply a relative residual-group readout; overlap or very weak signals can limit that inference.

Transverse nuclear magnetic resonance measures absorption from a radio-frequency magnetic field perpendicular to the static field. In the A phase, the resonance moved above the Larmor frequency; in the B phase it returned to that frequency despite the changed state. Anthony James Leggett connected the shift to collective spin-orbit structure. A predicted longitudinal resonance, driven parallel to the static field, supplied a further B-phase test.^[ref]

The helium-3 superfluid discovery experiment establishes why a new phase was inferred; comparing response channels then tests its internal structure. An unchanged peak in one geometry cannot, by itself, establish that nothing ordered.

The nuclear magneton is the unit of magnetic momentwhere $m_p$ is the proton mass. For a nuclear spin operator $\mathbf{I}$, the nuclear magnetic moment is $\mathbf{\mu }=g_I{\mu }_N\mathbf{I}/\hslash$. Consequently its gyromagnetic ratio is ${\gamma }_I=g_I{\mu }_N/\hslash$, and its Larmor frequency in a uniform magnetic field of magnitude $B$ is $|{\gamma }_I|B$. The nuclear magneton is smaller than the Bohr magneton by the electron-to-proton mass ratio.

The primary numerical study follows spherical protostellar collapse. A small central core forms before most of the envelope reaches it and subsequently grows by accretion. The abstract also describes infrared reprocessing by the dusty envelope. Spherical symmetry and the chosen radiative and thermodynamic treatment are model assumptions; rotation, magnetic fields and non-spherical accretion require additional physics.

Atmospheric winds across a magnetic field can induce electric currents. Their Ohmic heating converts mechanical energy into heat, at local rate $q=\mathbf{J}\cdot \mathbf{E}=J^2/\sigma$ in the simplest resistive description. To maintain hot-Jupiter radius inflation, current paths and deposited heat must influence the deep interior; dissipation restricted to optically thin upper layers need not maintain the planet's entropy.