Magnetic Field: Spatial Curl and Material Response

01From potential curl to spatial state

ASD first obtains magnetic interaction from a true-distance correction in a specified motion configuration. Magnetic field subsequently represents continuous spatial response. Given effective vector potential $\mathbf A_\tau$, define $\mathbf B=\nabla\times\mathbf A_\tau$. Curl measures local circulating variation and can describe torsion of shear orientation. The definition alone does not specify how a charge or needle responds.

In the source-free transverse effective sector with a specified gauge and compatible boundaries, let $\mathbf A_T=\mathbf A_\tau^\perp$ and $\mathbf A_T=N_T\mathbf u_T$:

$\mathbf u_T$ is a length-valued transverse spatial state, $\boldsymbol\Omega_T=\nabla\times\mathbf u_T$ a dimensionless local curl, and $N_T$ has units V·s/m². $\mathbf B$ is measured in teslas and $\mathbf E_T$ in V/m. $\boldsymbol\Omega_T$ describes spatial orientation change, not temporal angular velocity or a closed topological number; the radiation electric field represents the same state's process change.

Sourced regions still define magnetic field by potential curl, but directly mapping a general high-speed topological core to this spatial state requires a source equation. This page develops the established effective responses in the order of definition, constitutive coupling and variation.

02Quasistatic sources and magnetic representation

Let source and test-charge velocities be $\mathbf u,\mathbf v$ relative to the chosen background. Velocity $\mathbf u$ is distinct from displacement state $\mathbf u_T$. At lowest slow-source near-zone quasistatic order, $\mathbf A_\tau\simeq\Phi_{\mathrm{static}}\mathbf u/c^2$. With spatially constant source velocity, the curl product identity and $\mathbf E_{\mathrm{static}}=-\nabla\Phi_{\mathrm{static}}$ yield:

This field represents source-motion response rather than deriving magnetic origin through a frame transformation. Varying $\mathbf u$ adds $\Phi_{\mathrm{static}}\nabla\times\mathbf u/c^2$; time dependence also contributes to the electric field.

03Deriving matter response from the action

Within the effective source domain use $L=L_0-q\Phi_\tau+q\mathbf v\cdot\mathbf A_\tau$, with $L_0$ reconstructed from ASD momentum–energy. Test charge is $q$, mechanical momentum $\mathbf p=m_0\gamma\mathbf v$, and $t$ the same process parameter. Manuscript equation (3.21) gives the velocity and position derivatives:

$A_i$ is a component of $\mathbf A_\tau$, and repeated $j$ is summed. Along the trajectory $dA_i/dt=\partial_tA_i+v_j\partial_jA_i$. Substitution into Euler–Lagrange and rearranging vector-potential derivatives gives:

The antisymmetric gradient satisfies $v_j(\partial_iA_j-\partial_jA_i)=[\mathbf v\times(\nabla\times\mathbf A_\tau)]_i$, producing the magnetic term. Canonical momentum is $\mathbf P=\mathbf p+q\mathbf A_\tau$, while mechanical momentum remains $\mathbf p$. The Lorentz-force form is a variational consequence of the selected minimal coupling, not a hidden premise of the earlier LVE derivation.

Insert the quasistatic source field into $q\mathbf v\times\mathbf B$ and expand the triple product:

For common velocity perpendicular to $\mathbf E_{\mathrm{static}}$, the retained second-order magnetic term is $-\beta^2q\mathbf E_{\mathrm{static}}$, reproducing the leading LVE correction. The general two-velocity direction structure uses both minimal coupling and quasistatic source potential. A second-order total-force comparison must also retain electric corrections at that order.

With $E_f=\mathbf v\cdot\mathbf p-L_0$ and no explicit time dependence of the free term, work–energy gives:

The magnetic term is perpendicular to velocity and changes momentum direction without directly changing free motion energy in this model. A point charge at rest relative to the field representation, without a magnetic moment, has zero magnetic force; a time-dependent electric field can still act on it. A stationary magnetic needle has an internal moment and requires moment coupling, rather than the moment-free point-charge argument.

04Currents, wires and magnetic moments

Macroscopic current is directed transport of several charged constituents. Let $\rho_a$ be signed charge density and $\mathbf u_a$ velocity of constituent $a$. Define:

Positive and negative densities may cancel in $\rho_e$ while differing velocities leave $\mathbf J$ nonzero. For localized, steady divergence-free currents and decaying boundaries, take the curl of the preceding page's linear kernel. Let $\mathbf r'$ be source position, $\mathbf R=\mathbf r-\mathbf r'$, $R=|\mathbf R|$. Using $\nabla(1/R)=-\mathbf R/R^3$ gives:

For a thin wire replace $\mathbf J\,d^3r'$ by $I\,d\boldsymbol\ell'$ to obtain the Biot–Savart line element. With $\nabla^2(1/R)=-4\pi\delta^3(\mathbf R)$ and a divergence-free potential, one obtains $-\nabla^2\mathbf A_\tau=\mu_0\mathbf J$ and $\nabla\times\mathbf B=\mu_0\mathbf J$. For parallel straight wires much longer than their separation, neglecting end effects, symmetry and effective material response give:

$d$ is wire separation, $I_1,I_2$ are currents along chosen directions, and $L$ is the affected length. Magnitudes use absolute current values; direction follows current crossed with field: same-direction currents attract and opposite currents repel. This is reproduction within effective equations and coupling. Microscopic correspondence must include electrons, ions, cross-interactions, line densities and boundaries, not a simple sum of the co-moving two-charge special case.

Permanent magnets also involve average microscopic moment orientation. Manuscript equation (3.32) writes $\mathbf M=\sum_a n_a\langle\boldsymbol\mu_a\rangle$, with number density $n_a$ and moment $\boldsymbol\mu_a$. Equivalent bound volume and surface currents are $\mathbf J_b=\nabla\times\mathbf M$ and $\mathbf K_b=\mathbf M\times\hat{\mathbf n}$, where $\hat{\mathbf n}$ is the outward normal. Surface current $\mathbf K_b$ is not the LVE length factor.

For a dipole small relative to external-field variation, with a fixed moment parameter, equation (3.33) gives $U_\mu=-\boldsymbol\mu\cdot\mathbf B$, $\boldsymbol\tau=\boldsymbol\mu\times\mathbf B$, and $\mathbf F_\mu=\nabla(\boldsymbol\mu\cdot\mathbf B)$. Uniform fields exert torque; nonuniform fields can also exert net force. Induced moments require the material response. Torque $\boldsymbol\tau$ is distinct from the potential subscript $\tau$. Microscopic intrinsic moments, spin and gyromagnetic ratios remain stable-structure problems.

05Induction connects changing fields and material circuits

Taking curl of the smooth potential definitions removes the scalar gradient:

These are geometrical identities of the potential representation. Stokes' theorem for fixed boundary $C$ and fixed surface $S$ gives electromotive force $\mathcal E=\oint_C\mathbf E\cdot d\boldsymbol\ell=-d\int_S\mathbf B\cdot d\mathbf S/dt$. Changing transverse orientation connects to circulating electric response.

For a smooth closed circuit moving with the same material, write local circuit velocity $\mathbf w$. Effective force per charge includes $\mathbf E+\mathbf w\times\mathbf B$. The moving-surface transport identity gives manuscript equation (3.36):

A fixed circuit recovers the preceding relation; sliding contacts, disconnection or reconnection require the actual charge path. Induction identities alone do not determine speed. The next page obtains free propagation from transverse action and restoring coefficients.

06What vacuum coefficients and material response describe

Charged structures and accompanying fields in a material can shift or deform relative to one another; internal magnetic moments and closed orientations can respond collectively. Manuscript (3.47h) therefore organizes material response as

Here $\mathbf D$ is electric displacement, $\mathbf P_{\rm pol}$ is polarization rather than canonical momentum $\mathbf P$, and $\mathbf M$ is magnetization. The auxiliary field $\mathbf H$ subtracts the magnetization contribution. Only in a local, linear, isotropic approximation without magnetoelectric cross coupling does one further write $\mathbf D=\varepsilon\mathbf E$ and $\mathbf B=\mu\mathbf H$. Vacuum $\varepsilon_0,\mu_0$ and material $\varepsilon,\mu$ belong to different levels; a common interaction origin does not require every material to satisfy $\varepsilon\mu=c^{-2}$.

This connects ASD’s spatial account to material readings: polarization and magnetization record adjustments of different internal freedoms. A tensor-valued effective constitutive law alone does not establish a change in vacuum constants. The relation between static coupling, free-transverse inertia and restoring coefficients is developed on the electromagnetic-wave page.