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A particle entering a uniform axial magnetic field on the tube axis follows a transverse circle through that entry point. Its guiding centre is one Larmor radius from the axis, so the maximum radial displacement from the axis is twice the Larmor radius. For tube radius $a$, clearance requires $v_{\perp }\le |q|Ba/(2m)$ when the transit samples a complete gyration. If the transit is shorter, the actual sampled arc determines clearance instead. The axis is revisited after each full cyclotron period, not after each half-period.

A circular loop of radius $R$ carrying electric current $I$ has, on its axis,where the sign of $\hat{z}$ is fixed by the right-hand rule. At its centre, $B(0)={\mu }_{0}I/(2R)$.

Put the loop in the $xy$ plane, centered at the origin, with positive electric current counterclockwise when viewed from $+z$. The right-hand rule then gives the central magnetic field along $+z$. Parameterize source positions by the angle $\varphi$ and observe at $(0,0,z)$:Here $R$ without bold type is the loop radius, whereas $\mathbf{R}$ is the source-to-observer displacement. Their cross product isThe denominator $|\mathbf{R}{|}^3=(R^2+z^2{)}^{3/2}$ is constant around the loop. The integrals of sine and cosine over a full turn vanish, while the axial contribution adds to $2\pi R^2$. Substitution into the Biot-Savart law proves the on-axis formula. Symmetry cancels transverse components; it does not cancel the axial components.

For an original one-turn example in the same vacuum approximation, choose $R=0.04\mathrm{m}$ and $I=2\mathrm{A}$. With ${\mu }_0\approx 4\pi \times 10^{-7}\mathrm{T}m/A$,At $z=0.03\mathrm{m}$, the source-to-observer distance is $0.05\mathrm{m}$, so $B(z)/B(0)=64/125=0.512$ and $B(z)\approx 16.08\mu \mathrm{T}$. At $z=-0.03\mathrm{m}$, the magnetic field has the same magnitude and still points along $+z$. The point has crossed the loop plane, but the loop's electric current has not reversed. Reversing the electric current reverses the magnetic field on both sides.

For $|z|\ll R$, a Taylor expansion gives $B(z)/B(0)=1-3z^2/(2R^2)+O(z^4/R^4)$: the center is a smooth maximum, with zero first derivative. For $|z|\gg R$,This is the axial magnetic dipole field of magnetic dipole moment $\mathbf{m}=I\pi R^2\hat{\mathbf{z}}$.^[ref] A compact closed loop has an inverse-cube axial far field, in contrast to the inverse-distance field of the infinite straight wire. The SI units of the exact loop formula are ${\mu }_{0}I{R}^2/(R^2+z^2{)}^{3/2}\sim {\mu }_{0}I/\text{length}$, hence tesla.

For $N$ coincident turns carrying the same electric current, the superposition principle multiplies the ideal result by $N$. A real coil has radial width and axial length: if those are not negligible, integrate or sum over its individual turns. This is why a single measured coil diameter and a turn count may not determine the full spatial magnetic field precisely. A collection of loops becomes a solenoid; integrating their axial contributions gives the on-axis magnetic field of a finite solenoid.

Model a finite solenoid as a continuum of coaxial turns shaped as circles of radius $a>0$, uniformly distributed over length $\ell$ with solenoid turn density $n=N/\ell$. Put its centre at $z=0$, so its ends are $z=\pm \ell /2$. With the same electric current orientation as in the long-solenoid approximation, its axial magnetic field isThis formula applies only on the axis, where symmetry cancels the transverse magnetic field. It does not describe points off the axis. It is exact for the continuum circular-turn model, which neglects winding thickness, discrete wire spacing, and helical advance. ^[ref].

The derivation is an especially direct use of the superposition principle. The on-axis magnetic field of a circular current loop at source position $z^{\prime }$ is ${\mu }_{0}I{a}^2/[2(a^2+(z-z^{\prime }{)}^2{)}^{3/2}]$. There are $nd{z}^{\prime }$ turns in an axial slice, soSubstitute $u=z-z^{\prime }$ and use the antiderivative $u/[a^2\sqrt{a^2+u^2}]$ to obtain the displayed expression. If the centre is instead at $z=z_0$, replace every $z$ in that expression by $z-z_0$.

At the centre and at either end, respectively,For aspect ratios $\ell /a=2,6,20$, the centre ratios are approximately $0.7071,0.9487,0.9950$. The end ratio tends to $1/2$ as $\ell /a$ becomes large, even though the centre ratio tends to $1$. Thus a long coil can have an excellent central magnetic field while the ends still differ strongly from that value. Convergence to the long-solenoid approximation is not uniform across the end region.

For a concrete geometry, give the earlier coil radius $a=0.025\mathrm{m}$, so $\ell /a=20$. Its central magnetic field is approximately $1.2504\mathrm{mT}$, its end value is $0.6275\mathrm{mT}$, and at $z=\ell$ its axial magnetic field is still $0.00277\mathrm{mT}$. The last point is outside the coil: finite length does not permit the zero-exterior conclusion of the infinite model. Off the axis, the magnetic field lines bend around to form the return field. A central axial value alone also does not determine the total magnetic flux through every turn of a short coil.

For a semiclassical closed electron orbit in a plane perpendicular to a uniform magnetic field, its wavevector-space area obeys $S_k(E_n)=2\pi eB(n+\gamma )/\hslash$. An ordinary parabolic band without an extra geometric phase has $\gamma =1/2$. The semiclassical equations rotate and scale the real-space orbit by $\hslash /(eB)$, giving $S_r=[\hslash /(eB){]}^2{S}_k$. For a parabolic band this is equivalent to $E_n=\hslash {\omega }_c(n+1/2)$.

Optical molasses surrounds slow chemical atoms with opposing red-detuned laser beams, usually three pairs along perpendicular axes. In the elementary Doppler cooling picture, each pair damps one component of velocity. The result resembles viscous drag, even though the atoms are moving through a vacuum rather than a liquid.

Uniform balanced optical molasses supplies velocity damping, without a restoring force towards a chosen position. An atom at rest has zero mean force wherever it sits within the uniform illuminated region. Without noise or external forces, $m\dot{v}=-\alpha v$ givesThe atom approaches a position determined by its initial condition, not the beam intersection. Recoil-driven diffusion and gravity eventually remove atoms from finite beams. A long residence time in optical molasses is therefore compatible with the absence of position trapping. An optical trap provides spatial confinement by a different force mechanism; a magneto-optical trap adds a position-dependent magnetic field and suitable beam electromagnetic polarization to create a restoring force.

Optical nonreciprocity means that exchanging source and detector paths need not restore the corresponding transmission when external conditions, including the magnetic field, are held fixed. The Faraday effect gives a linear polarization rotation with the same laboratory sense on both traversals; see Faraday rotation on a round trip.

Circular polarization selects the sign of the absorbed photon's angular momentum along the beam. Choose that direction as $z$ and define ${\sigma }^{+}$ by the absorption rule $m^{\prime }=m+1$, where $m$ is the magnetic quantum number. Spontaneous emission can return an atom to a different magnetic sublevel, so absorption followed by emission is a population-transfer cycle. A sublevel that the chosen beam cannot excite is dark to that beam. Darkness is relative to the selected transition and electromagnetic polarization, rather than a claim that the atom cannot interact with any light.

A static magnetic field along $z$ defines a stable axis and gives Zeeman splitting. A transverse oscillating magnetic field can redistribute magnetic sublevel populations when its frequency matches an allowed splitting. With the appropriate optical transition, that redistribution restores photon absorption: transmitted intensity decreases, or collected fluorescence increases. These are alternative readouts whose detailed sensitivity depends on the transition, viewing direction and electromagnetic polarization.

For propagation perpendicular to an equilibrium magnetic field, an ordinary plasma wave has its electric field parallel to that magnetic field, so the particle response involves no magnetic deflection. With immobile ions, the dispersion relation is ${\omega }^2={\omega }_{pe}^2+c^2{k}^2$. It is a transverse electromagnetic wave, despite its parallel particle motion.

A nearby compass needle changes direction when a wire carries an electric current. Reversing the current or exchanging above/below positions reverses the magnetic action, connecting electric currents with magnetic fields. The long-wire idealization has $B={\mu }_{0}I/(2\pi r)$ and circular orientation given by the right-hand rule.

For a light nonmagnetic rod of length $l$ wound uniformly with $N$ thin turns of mean radius $a$, the magnetic dipole moment is $m_B=NI\pi a^2$. Wire of cross-sectional area $A_w$ and density $\rho$ has mass approximately $M=2\pi aN\rho A_w$. The transverse central moment of inertia is $J=M(l^2/12+a^2/2)$ in the thin-winding approximation. Consequently the oscillation period in a transverse magnetic field does not depend on the number of turns: wire inertia and magnetic moment both scale with $N$.

An instrument that records a time-varying electrical signal.

A physicist.^[ref]

The Stern-Gerlach experiment with Walther Gerlach tested the deflection of a neutral silver atomic beam in a nonuniform magnetic field.

Study of ancient magnetic fields recorded in rocks. Correlation to geomagnetic polarity chronology can constrain the age of archaeological and fossil-bearing strata.

Test whether normal- and reversed-polarity remanent magnetization directions are statistically antipodal after suitable structural correction. Agreement supports recording of opposite polarities of the same large-scale magnetic field; failure can reveal overprints or other complications. Passing is not, by itself, proof of primary acquisition.

Study of ancient magnetic-field records preserved in rocks. Primary magnetizations can constrain former latitude and rotation, subject to age and alteration tests.

Two broad perfect conductors of width $w$, separated by gap $d\ll w$, carry opposite steady electric currents $I$ and connect through a terminating resistor $R$. Away from edges, Ampère's circuital law and the superposition principle give gap magnetic field magnitude ${\mu }_{0}I/w$. Each conductor is an equipotential, and Ohm's law sets their electric potential difference to $IR$. Planar symmetry and charge-free Gauss's law then give uniform gap electric field magnitude $IR/d$. Injection parallel to the upper current makes the electric and magnetic Lorentz forces opposite. Their cancellation selects speed $Rw/({\mu }_{0}d)$ independent of nonzero electric charge and of operating current; nonrelativistic kinetic energy is $m[Rw/({\mu }_{0}d){]}^2/2$.

In a planar magnetohydrodynamic shock with both velocity and magnetic field normal to the front, the normal magnetic field is continuous. Its normal Maxwell stress tensor contribution is consequently the same on both sides, and the Poynting vector vanishes because $\mathbf{u}\times \mathbf{B}=0$. The remaining jump conditions are the ordinary Rankine-Hugoniot conditions for a perfect gas. This is the purely parallel branch; an oblique or switch-on shock has additional transverse fields and is not covered by this reduction.

The normal stress along the local magnetic field, evaluated using the random parallel velocity. At zero equilibrium mean flow, its linear perturbation is obtained from $m\int v_{\parallel }^2\delta f{d}^{3}v$ together with the perturbation of the velocity-space measure.

A material exhibiting paramagnetism. Its induced magnetization points along the applied magnetic field in the linear-response regime, giving positive magnetic susceptibility. Unlike a ferromagnet, it has no spontaneous ordered moment in that phase. The inverse-temperature response of independent localized moments is one model, rather than a definition that includes all microscopic mechanisms of paramagnetism.