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The magnetic tension stress transports axial angular momentum along a poloidal magnetic field. In purely azimuthal ideal flow the poloidal energy flux is ${\mathbf{F}}_{E,p}=\Omega {\mathbf{F}}_{J,p}$. This proportionality is valid even at vanishing flux, where an ordinary ratio is undefined. The coupled fluxes explain how a torsional Alfvén wave redistributes rotation.

Braking caused by induced currents whose magnetic force opposes motion. energy removed from mechanical motion can become resistive heat; the force depends on magnetic-field gradients or boundary crossing, resistance and inductance.

For reversible quasistatic magnetic circuits with flux linkages ${\lambda }_i$, currents $I_i$ and mechanical coordinate $q$, write $d{W}_B={\sum }_i{I}_{i}d{\lambda }_i-F_{q}dq$. The Legendre transform $W_B^{\prime }={\sum }_i{I}_i{\lambda }_i-W_B$ obeys $d{W}_B^{\prime }={\sum }_i{\lambda }_{i}d{I}_i+F_{q}dq$. Thus $F_q=(\partial W_B^{\prime }/\partial q{)}_{I_i}$, while $F_q=-(\partial W_B/\partial q{)}_{{\lambda }_i}$. In a linear reciprocal system $W_B^{\prime }=W_B=\frac{1}{2}{\mathbf{I}}^{\mathsf{T}}L(q)\mathbf{I}$ numerically, but the variables held fixed in their force derivatives differ. The coenergy formulation includes the effect of electrical sources maintaining the currents; it avoids reversing the sign of the magnetic force by differentiating stored energy under the wrong constraint.

If a rigid body's inclined magnetic dipole rotates at angular speed $\Omega$, its radiated power is $C{\Omega }^4$. Conservation of rotational energy $I{\Omega }^2/2$ then gives

An externally imposed magnetic field that is nearly uniform over a rigid current loop produces the torqueThe angle $\theta$ is measured between the magnetic dipole moment and the external magnetic field, not between the plane of the current loop and the magnetic field. The magnetic dipole potential energy uses a dot product, whereas the torque uses a cross product. The torque is largest when the magnetic dipole moment is perpendicular to the magnetic field; parallel alignment minimizes the potential energy.^[ref]

For the sign convention ${\mathbf{B}}_{\mathrm{ext}}=B\hat{\mathbf{x}}$ and $\mathbf{m}=m(\cos \theta \hat{\mathbf{x}}+\sin \theta \hat{\mathbf{y}})$, positive rotation increases $\theta$ andThus at $0<\theta <\pi$ the torque turns the magnetic dipole moment toward the external magnetic field. Both parallel and antiparallel alignment have zero torque, but they are respectively a stable equilibrium and an unstable equilibrium. With no damping, a freely released current loop can undergo oscillation around the stable equilibrium; settling requires damping.

In the twelve-turn example, take $B=0.020\mathrm{T}$ and $\theta =60°$. The predicted torque is ${\tau }_z=-2.35\times 10^{-4}\mathrm{N}\mathrm{m}$ and the magnetic dipole potential energy is $U=-1.36\times 10^{-4}\mathrm{J}$. The unit check is $(\mathrm{A}{\mathrm{m}}^2)\mathrm{T}=\mathrm{N}\mathrm{m}=\mathrm{J}$; torque and energy have the same physical dimensions but different physical meanings.

A short derivation starts with the magnetic force on a current-carrying wire, $d\mathbf{F}=Id\mathbf{r}\times {\mathbf{B}}_{\mathrm{ext}}$. In an exactly uniform magnetic field, the net force is $I({\oint }_{C}d\mathbf{r})\times {\mathbf{B}}_{\mathrm{ext}}=0$, but its torque need not vanish: opposite sides produce a turning couple. Integrating $\mathbf{r}\times d\mathbf{F}$ gives $\mathbf{m}\times {\mathbf{B}}_{\mathrm{ext}}$. Integrating $dU=-{\tau }_{z}d\theta$ at fixed $m$ and $B$ gives the orientation potential energy. This is the mechanical orientation potential energy for a maintained electric current, with the external source held fixed; the energy accounting for field and power supply is a separate problem. A varying external magnetic field requires corrections across a finite current loop, and can also give a net force.

Spatial variations in energy, including crystal defects, can trap a magnetic domain wall. This is magnetic domain-wall pinning. Increasing the applied magnetic field can release the wall into a collective jump.

Uniform perpendicular fields with $E^2<c^2{B}^2$ admit a subluminal Lorentz boost eliminating the electric field. A laboratory particle initially at rest follows a circle in the adapted frame and a stretched cycloid in the laboratory. The mean drift velocity is $\mathbf{E}\times \mathbf{B}/B^2$, while its laboratory kinetic energy is periodic rather than unbounded.

For constant electrical resistivity $\eta$ and magnetic permeability ${\mu }_0$, let $E_B={\int }_V{B}^2/(2{\mu }_0)dV$ and $\mathbf{J}=\nabla \times \mathbf{B}/{\mu }_0$. In a fixed volume with zero boundary energy flux, multiply the resistive induction equation by $\mathbf{B}/{\mu }_0$ and integrate by parts. This gives the displayed balance. The first term transfers energy between the magnetic field and the fluid; the second is nonpositive and represents Joule heating. The latter does not imply that the total magnetic energy must decrease if fluid motion continually supplies it.

In vacuum or an approximately nonmagnetizable plasma, the field energy per volume is $B^2/(2{\mu }_0)$. magnetic pressure has the same scale, but a magnetic field also exerts directional tension, so its stress is not universally an isotropic pressure. Material magnetization can require a different constitutive treatment.

The general ideal magnetic-energy balance has outward-surface contribution ${\mu }_0^{-1}\oint [(u\cdot B)B-B^{2}u]\cdot dS$ and volume contribution $-\int u\cdot (J\times B)dV$. The latter vanishes for a force-free magnetic field. For an axisymmetric boundary rotating tangentially, $u\cdot B=\Omega f(\psi )$ and the directed flux of a thin tube is $2\pi d\psi$. Pair its exit and entry endpoints to obtain the displayed energy injection, counting each boundary endpoint once. Equal endpoint angular velocities inject no magnetic energy.

The shell-integrated magnetic energy spectrum distributes magnetic energy over wavenumbers. With a solenoidal isotropic spectral tensor normalized to ${\hat{C}}_{ii}=2H(k)$ and the inverse Fourier transform measure $d^{3}k/(2\pi {)}^3$, one has $\mathbb{E}|\mathbf{B}{|}^2={\pi }^{-2}\int k^{2}H(k)dk$. In Gaussian units, $E_B(k)=k^{2}H(k)/(8{\pi }^3)$ integrates to the energy density $\mathbb{E}|\mathbf{B}{|}^2/(8\pi )$. A spectrum per unit mass or normalized to $B^2/2$ has different constant factors. Scale-local fluctuation estimates give $\delta B_l^2$ proportional to $k{E}_B(k)$ at $k\sim l^{-1}$ when the spectrum is normalized consistently.

For weak-field sodium D lines neglecting hyperfine structure, the smallest adjacent spectral spacing in the full electronic pattern is $s_E=2{\mu }_{B}B/3$. A drawing without a labelled energy axis can still estimate $B$ if it includes both unsplit D lines with known wavelengths: their separation calibrates $hc(1/{\lambda }_2-1/{\lambda }_1)$. Divide the drawn split spacing by the drawn unsplit separation before using that energy scale. Graphical line thickness, unequal drawn spacings and uncalibrated rendering limit precision; they do not justify exact numerical field values.

Instantaneously reversing a Hamiltonian $H(B)$ satisfying $H(-B)=-H(B)$ leaves populations unchanged but rewrites them as a canonical distribution with ${\beta }^{\prime }=-\beta$. When the partition function is unchanged, the free energy changes sign and the heat capacity is unchanged.

A magnetic flux quantum is the flux scaleassociated with a coherent degree of freedom of electric charge $q\ne 0$. Its Aharonov-Bohm phase on a loop enclosing magnetic flux $\Phi$ is $q\Phi /\hslash$; adding ${\Phi }_0$ changes that phase by one full turn.

For an electron, the magnitude of the charge is $e$, giving $h/e$. A conventional superconducting Cooper pair has charge magnitude $2e$, giving $h/(2e)$. The relevant charge must therefore be stated: a superconducting flux scale and a single-electron interference period differ by a factor of two.

In the magnetic translation algebra, translations along two lattice vectors commute when the enclosed unit-cell magnetic flux obeysThis is a condition on the algebra for the specified cell, not a claim that every externally applied magnetic flux must be quantized. The distinct magnetic flux spectrum of a quantum ring illustrates flux-periodic energy levels.

The magnetic part of the Lorentz force is perpendicular to the particle velocity. Its power is therefore zero, so with no other forces the kinetic energy remains constant. This remains true relativistically because $dE/dt=\mathbf{F}\cdot \mathbf{v}$; a magnetic force can change momentum direction without changing total energy.

Magnetic hysteresis means that a material's magnetization depends on its field history as well as the present applied field. A closed quasistatic magnetic cycle dissipates energy per unit volume equal to its $H$–$B$ loop area. It is relevant to remanence in permanent magnets and core losses in electrical transformers.

For slow variation of the field compared with the gyration period, the magnetic moment $m{v}_{\perp }^2/(2B)$ is approximately conserved. Thus transverse random energy scales with $B$, and remains unchanged only when the relevant field strength is unchanged.

Magnetic reconnection changes connectivity between magnetic field lines. It requires departure from smooth magnetic flux freezing, for example through finite resistivity or other nonideal terms concentrated in a current sheet. Ideal viscous magnetic relaxation dissipates mechanical energy while retaining frozen connectivity, so viscosity alone does not provide reconnection.

For thin parallel circular plates of radius $a$ and separation $L$, the quasistatic axial field $E_z(t)$ induces $H_{\varphi }={ϵ}_{0}r{\dot{E}}_z/2$. Integrating cycle-averaged vacuum energy densities for sinusoidal drive gives ${\overline{U}}_B/{\overline{U}}_E=(\omega a/c{)}^2/8$. The same fields have radial Poynting flux $S_r=-{ϵ}_{0}r{E}_z{\dot{E}}_z/2$, whose leading net inward power equals the time derivative of stored electric energy. The small ratio diagnoses the quasistatic regime.

A neutral chemical atom with a magnetic dipole moment can experience a force in a nonuniform magnetic field, despite having zero net electric charge. For an internal state that follows the local field adiabatically, its position-dependent Zeeman energy supplies an effective trapping potential energy.

A low-field-seeking state has increasing energy with field magnitude and can be confined near a field minimum. The atom must retain the trapped internal state: rapid changes of field direction or a vanishing splitting can permit transitions to untrapped states. The trap therefore selects both spatial motion and internal spin character. In atom-correlation experiments, releasing this trapped population provides a controlled source for time-of-flight atom detection.