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Maxwell stress tensor

The Maxwell stress tensor is a second-rank tensor in classical electromagnetism that quantifies the momentum flux and mechanical stresses exerted by electric and magnetic fields on matter, providing a framework to compute forces via surface integrals over field configurations.^[1] It arises from the electromagnetic field's ability to carry energy and momentum, analogous to stresses in a fluid or elastic medium, and is essential for understanding interactions without relying on direct action-at-a-distance between charges.^[2] Introduced by James Clerk Maxwell in his seminal 1873 work A Treatise on Electricity and Magnetism, the tensor formalized the mechanical properties of the electromagnetic field, building on earlier ideas from his 1865 paper "A Dynamical Theory of the Electromagnetic Field" where he first conceptualized fields as carrying stress.^[3] In the treatise, Maxwell derived it to describe how electric fields produce tensions along field lines and pressures perpendicular to them, with magnetic fields contributing oppositely, thus unifying disparate observations of attraction and repulsion in conductors and dielectrics.^[1] The classical expression for the tensor \mathbf{T} in Cartesian coordinates is given by

T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right),

where \epsilon_0 and \mu_0 are the vacuum permittivity and permeability, \mathbf{E} and \mathbf{B} are the electric and magnetic field vectors, E^2 = \mathbf{E} \cdot \mathbf{E}, B^2 = \mathbf{B} \cdot \mathbf{B}, and \delta_{ij} is the Kronecker delta; this form is symmetric in vacuum.^[1]^[2] Physically, the Maxwell stress tensor represents the electromagnetic field's contribution to the total stress-energy tensor, enabling the derivation of the Lorentz force density \mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} through its divergence, \nabla \cdot \mathbf{T} + \frac{\partial}{\partial t} (\epsilon_0 \mathbf{E} \times \mathbf{B}) = -\mathbf{f}, which ensures local conservation of momentum in the presence of charges and currents.^[1] It highlights the field's role as a mediator of forces, such as the attraction between parallel current-carrying wires or the pressure from electromagnetic radiation on surfaces.^[2] In applications, it is widely used to calculate net forces on objects like capacitors, solenoids, or antennas by integrating \oint \mathbf{T} \cdot d\mathbf{A} over a closed surface enclosing the object, avoiding explicit volume integrations of charge distributions.^[1] Beyond classical contexts, extensions appear in relativistic electrodynamics and magnetoelasticity, where it couples electromagnetic and mechanical stresses in materials.^[4]

Background and Motivation

Physical Interpretation

The Maxwell stress tensor quantifies the electromagnetic forces and stresses acting on materials, representing the force per unit area due to electric and magnetic fields interacting with charges or currents. In this context, electromagnetic stress emerges from the Lorentz force density, providing a framework to describe how fields exert traction or pressure on surfaces within or bounding matter. This tensor encapsulates the field's ability to transmit momentum, much like pressure in a fluid or tension in a solid. Analogous to the Cauchy stress tensor in continuum mechanics, the Maxwell stress tensor describes the flux of momentum through a surface in the electromagnetic field, enabling the total force on an enclosed volume to be computed via a surface integral over an arbitrary closed surface surrounding the object. This momentum flux perspective treats the electromagnetic field as carrying and transferring momentum, resolving the need to integrate volume forces directly and instead leveraging boundary conditions for practical calculations. The approach aligns with conservation laws, where the divergence of the tensor plus the time rate of change of the electromagnetic momentum density equals the negative of the Lorentz force density on matter. Representative physical effects illustrate its utility: radiation pressure from electromagnetic waves, such as light impinging on a mirror, arises as the tensor's normal component yields a net momentum transfer proportional to the wave's energy density. Forces on dielectrics in nonuniform electric fields, like the pull of a polarizable material toward regions of higher field strength, are captured by integrating the tensor over the material's surface, highlighting electrostatic attractions without explicit charge distributions. In magnetostatic configurations, the tensor accounts for repulsive or levitating forces, as seen in magnetic levitation systems where superconducting materials expel fields, creating upward thrusts balanced against gravity. A key conceptual role of the tensor is in addressing the "hidden momentum" paradox, where systems with crossed steady electric and magnetic fields appear to violate total momentum conservation during interactions; the tensor's surface integral reveals the field's hidden momentum contribution, ensuring overall balance without mechanical motion in the matter alone.

Historical Context

James Clerk Maxwell introduced the foundational concept of electromagnetic stresses in his 1873 Treatise on Electricity and Magnetism, where he described forces between charged bodies as arising from mechanical tensions and pressures within the luminiferous ether, thereby providing a field-based alternative to action at a distance. This approach modeled the ether as an elastic medium capable of transmitting electromagnetic disturbances, with stresses analogous to those in continuous media, enabling a unified treatment of electric and magnetic interactions. In the late 1880s and early 1890s, Oliver Heaviside advanced Maxwell's ideas by deriving explicit expressions for the forces and stresses in electromagnetic fields, culminating in his 1892 paper "On the Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field." Heaviside's formulation presented the stress components as a symmetric tensor, facilitating calculations of mechanical forces on conductors and dielectrics immersed in fields, and emphasized its role in conserving momentum across field and matter interactions.^[5] Hendrik Lorentz built upon these developments in the 1890s, integrating the stress tensor into his electron theory to account for electromagnetic momentum in moving systems. In his 1895 Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern, Lorentz demonstrated that the divergence of the stress tensor, combined with the field's momentum density, yields the total force on matter, ensuring conservation laws hold even for accelerated charges. His work refined the tensor for dynamic cases, linking field stresses to the motion of ponderable bodies without invoking ether drag. The early 20th century saw a profound evolution prompted by the 1887 Michelson-Morley experiment, which failed to detect Earth's motion through the ether and prompted its abandonment.^[6] Albert Einstein's 1905 special relativity eliminated the ether entirely, reinterpreting electromagnetic stresses as intrinsic properties of the field in inertial frames. Hermann Minkowski extended this in 1908 by embedding the Maxwell stress tensor within a relativistic four-dimensional framework as part of the electromagnetic stress-energy-momentum tensor, achieving full covariance under Lorentz transformations and paving the way for general relativity.^[7]

Mathematical Formulation

General Tensor Expression

The Maxwell stress tensor \mathbf{T} is a second-order symmetric tensor that quantifies the momentum flux density due to electromagnetic fields in vacuum, originally conceptualized by James Clerk Maxwell to describe stresses in the electromagnetic medium.^[3]^[8] It arises from the conservation of electromagnetic momentum and provides a framework for calculating forces on charges and currents without direct integration of the Lorentz force.^[8] In Cartesian coordinates, the general expression for the Maxwell stress tensor in SI units is given by

T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right) + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right),

where \epsilon_0 is the vacuum permittivity, \mu_0 is the vacuum permeability, \mathbf{E} = (E_x, E_y, E_z) is the electric field vector, \mathbf{B} = (B_x, B_y, B_z) is the magnetic field vector, E^2 = \mathbf{E} \cdot \mathbf{E}, B^2 = \mathbf{B} \cdot \mathbf{B}, and \delta_{ij} is the Kronecker delta (with \delta_{ij} = 1 if i = j and 0 otherwise).^[8] This form encapsulates both electric and magnetic contributions, applicable to time-varying fields.^[8] The tensor components distinguish between normal and shear stresses: the diagonal elements (T_{xx}, T_{yy}, T_{zz}) represent normal stresses along the principal axes, arising from the field magnitudes and indicative of tension or compression, while the off-diagonal elements (T_{xy}, T_{xz}, etc.) capture shear stresses due to field directionality.^[8] For instance, T_{xx} = \epsilon_0 \left( E_x^2 - \frac{1}{2} (E_y^2 + E_z^2) \right) + \frac{1}{\mu_0} \left( B_x^2 - \frac{1}{2} (B_y^2 + B_z^2) \right), highlighting how aligned fields produce tensile effects parallel to the field lines.^[8] The units of T_{ij} are those of stress, equivalent to pascals (Pa) or newtons per square meter (N/m²), as it measures force per unit area transmitted by the fields.^[8] Regarding sign convention, positive diagonal components typically denote tensile stress (pulling the medium apart along field lines), while negative values indicate compressive stress perpendicular to them, consistent with the tensor's role in balancing electromagnetic momentum flow.^[8] This convention aligns with the negative sign in the momentum conservation equation relating to field momentum density.^[8]

Derivation from Maxwell's Equations

The derivation of the Maxwell stress tensor arises from the need to express the conservation of momentum in electromagnetic fields, extending the principles underlying Poynting's theorem for energy conservation. Poynting's theorem states that the rate of change of electromagnetic energy density plus the divergence of the Poynting vector equals the negative of the power delivered to matter by the fields:

\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E},

where u = \frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 is the energy density and \mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B} is the Poynting vector representing energy flux.^[9] Analogously, electromagnetic fields carry momentum, with the momentum density given by \mathbf{g} = \epsilon_0 \mathbf{E} \times \mathbf{B} = \frac{1}{c^2} \mathbf{S}, where c = 1/\sqrt{\epsilon_0 \mu_0} is the speed of light; this relation follows from the relativistic structure of the fields but holds in the non-relativistic limit for momentum transport.^[10]^[9] The local conservation law for electromagnetic momentum takes the form

\frac{\partial g_i}{\partial t} + \frac{\partial T_{ij}}{\partial x_j} = -f_i,

where T_{ij} is the i-component of momentum flux across a surface normal to the j-direction (the Maxwell stress tensor), and f_i is the i-component of the force density exerted by the fields on matter. This equation expresses that the rate of change of field momentum density plus the net flux of momentum out of a volume equals the negative of the force on the matter within that volume, ensuring overall momentum conservation. The force density on matter is the Lorentz force density, \mathbf{f} = \rho \mathbf{E} + \mathbf{J} \times \mathbf{B}, where \rho is the charge density and \mathbf{J} is the current density.^[10]^[9] To derive the explicit form of the divergence \partial T_{ij}/\partial x_j, substitute expressions for \rho and \mathbf{J} from Maxwell's equations into the Lorentz force density: \rho = \epsilon_0 \nabla \cdot \mathbf{E} from Gauss's law, and \mathbf{J} = \frac{1}{\mu_0} \nabla \times \mathbf{B} - \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} from Ampère's law with Maxwell's correction. This yields

\mathbf{f} = \epsilon_0 (\nabla \cdot \mathbf{E}) \mathbf{E} + \frac{1}{\mu_0} (\nabla \times \mathbf{B}) \times \mathbf{B} - \epsilon_0 \left( \frac{\partial \mathbf{E}}{\partial t} \right) \times \mathbf{B}.

For symmetry and to incorporate the remaining equations, add the vanishing term \frac{1}{\mu_0} (\nabla \cdot \mathbf{B}) \mathbf{B} (since \nabla \cdot \mathbf{B} = 0 from Gauss's law for magnetism), giving

\mathbf{f} = \epsilon_0 (\nabla \cdot \mathbf{E}) \mathbf{E} + \frac{1}{\mu_0} (\nabla \cdot \mathbf{B}) \mathbf{B} + \frac{1}{\mu_0} (\nabla \times \mathbf{B}) \times \mathbf{B} - \epsilon_0 \left( \frac{\partial \mathbf{E}}{\partial t} \right) \times \mathbf{B}.

^[10] Next, compute the time derivative of the momentum density:

\frac{\partial \mathbf{g}}{\partial t} = \epsilon_0 \left( \frac{\partial \mathbf{E}}{\partial t} \times \mathbf{B} + \mathbf{E} \times \frac{\partial \mathbf{B}}{\partial t} \right).

Substitute Faraday's law \nabla \times \mathbf{E} = -\partial \mathbf{B}/\partial t, so \mathbf{E} \times \partial \mathbf{B}/\partial t = \mathbf{E} \times (-\nabla \times \mathbf{E}) = (\nabla \times \mathbf{E}) \times \mathbf{E}. The -\epsilon_0 (\partial \mathbf{E}/\partial t) \times \mathbf{B} term in \mathbf{f} cancels directly with the \epsilon_0 (\partial \mathbf{E}/\partial t \times \mathbf{B}) term in \partial \mathbf{g}/\partial t, leaving \epsilon_0 \mathbf{E} \times (\partial \mathbf{B}/\partial t) = \epsilon_0 (\nabla \times \mathbf{E}) \times \mathbf{E} from this contribution.^[10]^[9] Combining all terms, \mathbf{f} + \partial \mathbf{g}/\partial t now consists of field derivatives that can be rewritten using vector identities, such as the identity for \mathbf{A} \times (\nabla \times \mathbf{A}) = \frac{1}{2} \nabla (A^2) - (\mathbf{A} \cdot \nabla) \mathbf{A} applied separately to electric and magnetic contributions (with appropriate factors of \epsilon_0 and $1/\mu_0). These manipulations express \mathbf{f} + \partial \mathbf{g}/\partial t as the divergence of a second-rank tensor \mathbf{T}:

\mathbf{f} + \frac{\partial \mathbf{g}}{\partial t} = \nabla \cdot \mathbf{T},

or equivalently, \nabla \cdot \mathbf{T} = -\mathbf{f} - \partial \mathbf{g}/\partial t. This demonstrates that the Maxwell stress tensor T_{ij} encapsulates the momentum flux due to electromagnetic stresses, mediating forces between fields and matter through field interactions rather than direct contact. The derivation relies on all four Maxwell's equations and confirms the tensor's role in conserving total momentum.^[10]^[9]

Applications in Electromagnetic Fields

Electrostatic Limit

In the electrostatic limit, where magnetic fields vanish (\mathbf{B} = \mathbf{0}), the Maxwell stress tensor reduces to a form that captures the mechanical stresses arising solely from electric fields.^[11] The tensor components are

T_{ij} = \epsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2 \right),

where \epsilon_0 is the vacuum permittivity, \mathbf{E} is the electric field vector, E^2 = \mathbf{E} \cdot \mathbf{E}, and \delta_{ij} is the Kronecker delta.^[8] This expression reflects the anisotropic nature of electrostatic stresses: tensile along field lines and compressive perpendicular to them.^[8] The total electrostatic force \mathbf{F} on a charged object or assembly is computed by integrating the tensor over a closed surface S enclosing the object:

\mathbf{F} = \oint_S \mathbf{T} \cdot d\mathbf{A},

where d\mathbf{A} is the outward-pointing area element.^[12] This surface integral equals the volume integral of \nabla \cdot \mathbf{T} inside S, which balances the Lorentz force density on charges within, ensuring momentum conservation.^[12] A representative application is the force between parallel-plate capacitor plates, each of area A and separated by distance d \ll \sqrt{A}, with uniform surface charge densities \pm \sigma. The electric field between the plates is E = \sigma / \epsilon_0, directed normally from the positive to negative plate. The relevant stress tensor component normal to a plate is T_{nn} = \frac{1}{2} \epsilon_0 E^2, yielding an attractive force magnitude F = \frac{1}{2} \epsilon_0 E^2 A = \frac{\sigma^2 A}{2 \epsilon_0} per plate pair.^[8] This result matches the energy-based derivation F = -\frac{\partial U}{\partial d}, where U = \frac{1}{2} \epsilon_0 E^2 (A d) is the field energy.^[8] Another key example involves a dielectric slab of thickness l, dielectric constant \kappa > 1, and cross-sectional area matching the capacitor plates, partially inserted into the uniform field between fixed-voltage plates. The slab experiences a net force pulling it fully into the capacitor, computed via the stress tensor as F = \frac{1}{2} \epsilon_0 (\kappa - 1) l w E_0^2, where w is the slab width perpendicular to both the field \mathbf{E_0} and insertion direction, and E_0 = V/d with voltage V.^[13] This force originates at the dielectric-vacuum interfaces, where the field discontinuity induces bound surface charges via polarization \mathbf{P} = \epsilon_0 (\kappa - 1) \mathbf{E}; depolarization effects reduce the internal field to E = E_0 / \kappa, enhancing the stress imbalance that drives the attraction.^[13] The tensor also reveals behaviors such as repulsive forces between like-charged parts of a conductor. For a uniformly charged conducting sphere of total charge q and radius a, divided into two hemispheres along the equator, the stress tensor predicts a repulsive force of magnitude F = \frac{q^2}{32 \pi \epsilon_0 a^2} between the hemispheres, arising from the outward normal stress across the equatorial plane where the field exerts a push despite being zero inside the conductor.^[12]

Magnetostatic Limit

In the magnetostatic limit, where electric fields vanish (\mathbf{E} = \mathbf{0}) and magnetic fields are static, the Maxwell stress tensor reduces to its purely magnetic form, which describes the mechanical stresses arising from magnetic interactions on currents and magnetic materials.^[14] This specialization eliminates electric contributions, focusing on how magnetic fields exert forces analogous to tension and pressure in materials.^[15] The explicit expression for the tensor components is

T_{ij} = \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right),

where \mathbf{B} is the magnetic field vector, \mu_0 is the permeability of free space, and \delta_{ij} is the Kronecker delta.^[14] The off-diagonal elements T_{ij} (for i \neq j) represent shear stresses, while the diagonal elements capture normal stresses, including a magnetic pressure term B^2 / (2 \mu_0) that acts isotropically outward and a tension along field lines.^[15] The total force \mathbf{F} on a volume V enclosing currents or magnetic materials is obtained via the surface integral over the boundary S:

\mathbf{F} = \oint_S \mathbf{T} \cdot d\mathbf{A},

which, by the divergence theorem, equals \int_V (\nabla \cdot \mathbf{T}) \, dV. In magnetostatics, \nabla \cdot \mathbf{T} = \mathbf{j} \times \mathbf{B}, matching the Lorentz force density and enabling computation without direct knowledge of internal currents.^[14] For a straight current-carrying wire of length L in a uniform external field \mathbf{B}, integrating \mathbf{T} over a cylindrical Gaussian surface around the wire yields the force \mathbf{F} = I L \hat{\mathbf{l}} \times \mathbf{B}, where I is the current and \hat{\mathbf{l}} the wire direction; this highlights the tensor's utility in resolving forces on extended conductors.^[16] A practical example is the tension in a circular current loop due to its self-generated field, where the hoop stress from T_{\theta\theta} \approx -B^2 / (2 \mu_0) (along the azimuthal direction) balances the magnetic pressure, providing tension to prevent radial expansion under the outward Lorentz forces.^[14] Similarly, for a ferromagnetic core partially inserted into a solenoid, the tensor integrated over the core's end faces reveals an attractive force pulling the core inward, driven by the magnetic pressure difference B^2 / (2 \mu_0) across the air-core interface, with typical values on the order of several newtons for laboratory solenoids carrying amperes.^[17] This approach underscores the tensor's role in engineering applications like actuators. The Maxwell stress tensor also addresses inconsistencies in Ampère's original force law between current elements, which failed to satisfy Newton's third law due to neglecting momentum carried by the electromagnetic field; by incorporating field momentum flux, the tensor ensures overall conservation.^[18]

Properties and Extensions

Eigenvalues and Principal Axes

The eigenvalues of the Maxwell stress tensor T_{ij} are determined by solving the characteristic equation \det(T_{ij} - \lambda \delta_{ij}) = 0, where the solutions \lambda correspond to the principal stresses, representing the maximum and minimum normal stresses exerted by the electromagnetic field on a surface perpendicular to the principal axes.^[19] These principal values and directions provide insight into the directional nature of electromagnetic forces, analogous to principal stresses in continuum mechanics. In electrostatics, the eigenvalues relate directly to the electric field intensity along the principal axes: the tensor can be expressed as T_{ij} = \epsilon_0 (E_i E_j - \frac{1}{2} \delta_{ij} E^2), leading to one positive eigenvalue along the field direction indicative of tension and two negative eigenvalues perpendicular to it, signifying compression.^[20] Similarly, in magnetostatics, the form T_{ij} = \frac{1}{\mu_0} (B_i B_j - \frac{1}{2} \delta_{ij} B^2) yields eigenvalues reflecting magnetic tension along field lines and pressure transverse to them, with the principal axes aligned with the magnetic field orientation.^[21] A representative example occurs for a uniform electric field \mathbf{E} = E \hat{z}, where the energy density is u = \frac{1}{2} \epsilon_0 E^2. The eigenvalues are \lambda_1 = u (tensile, along the field) and \lambda_2 = \lambda_3 = -u (compressive, perpendicular), with principal axes parallel and orthogonal to \mathbf{E}.^[19] The trace of the tensor is T_{ii} = -u, so the sum of the eigenvalues equals -u; however, the deviatoric (traceless) part of the tensor, obtained by subtracting the isotropic pressure contribution p = -u/3, has eigenvalues summing to zero, thereby balancing the tensile and compressive components across the principal directions.^[20]

Relativistic Generalization

The relativistic generalization of the Maxwell stress tensor arises within the framework of special relativity, where electromagnetic phenomena are described using four-dimensional spacetime tensors to ensure Lorentz covariance. This extension embeds the three-dimensional Maxwell stress tensor into the broader electromagnetic stress-energy tensor, which accounts for both energy and momentum densities in a unified manner. The concept was first introduced by Hermann Minkowski in his 1908 paper on the fundamental equations for electromagnetic processes in moving bodies.^[7] The covariant form of the electromagnetic stress-energy tensor is given by

T^{\mu\nu} = F^{\mu\alpha} F^\nu{}_\alpha - \frac{1}{4} g^{\mu\nu} F_{\alpha\beta} F^{\alpha\beta},

where F^{\mu\nu} is the electromagnetic field tensor, g^{\mu\nu} is the Minkowski metric tensor (with signature (+,-,-,-) or (-,+,+,+) depending on convention), and the expression is symmetric in its indices.^[22] This tensor encapsulates the energy-momentum content of the electromagnetic field in vacuum, derived from the Lagrangian density of the field. In the laboratory frame, the spatial components T^{ij} (for i,j = 1,2,3) recover the familiar three-dimensional Maxwell stress tensor, describing the momentum flux due to electromagnetic forces, while the time-time component T^{00} corresponds to the electromagnetic energy density u = \frac{1}{2} (\epsilon_0 E^2 + \frac{1}{\mu_0} B^2), and the mixed time-space components T^{0i} yield the Poynting vector components representing the energy flux or momentum density.^[23] A key feature of this relativistic formulation is the conservation law \partial_\mu T^{\mu\nu} = -f^\nu, where f^\nu = F^{\nu\lambda} j_\lambda is the four-force density exerted by the field on charges and currents described by the four-current j^\lambda. This equation, derived directly from the covariant Maxwell equations \partial_\mu F^{\mu\nu} = \mu_0 j^\nu and \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0, unifies the conservation of energy and momentum in relativistic electromagnetism, extending beyond non-relativistic static cases to handle dynamic fields and observer-dependent transformations.^[22] In the absence of sources (j^\nu = 0), the tensor is divergence-free, \partial_\mu T^{\mu\nu} = 0, reflecting the self-conservation of the electromagnetic field's energy-momentum.^[24]

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### Maxwell Stress Tensor: Definition, History, Key Components, and Mathematical Formulation
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