9.6 Electromagnetic stress tensor

Electromagnetic stress tensor

The Maxwell stress tensor provides a way to calculate the total electromagnetic force on an object without tracking every charge and current inside it. Instead of summing up all the microscopic Lorentz forces, you integrate the stress tensor over a closed surface surrounding the object. This makes it indispensable for problems involving forces on dielectrics, conductors, and plasmas, and it connects directly to the conservation of field momentum covered earlier in this unit.

Definition and components

The Maxwell stress tensor in vacuum is a symmetric rank-2 tensor (a 3×3 matrix in the non-relativistic setting) defined component-wise as:

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

where $i, j = 1, 2, 3$ label spatial directions, and $\delta_{ij}$ is the Kronecker delta.

Each component $T_{ij}$ represents the flux of the $i$-th component of electromagnetic momentum through a surface whose normal points in the $j$-direction. In more physical terms:

• Diagonal components ($T_{ii}$) give the normal stress (pressure) the field exerts on a surface perpendicular to the $i$-axis.
• Off-diagonal components ($T_{ij}$, $i \neq j$) give the shear stress, the tangential force per unit area the field exerts on that surface.

The tensor is symmetric: $T_{ij} = T_{ji}$. This follows from conservation of angular momentum for the electromagnetic field (assuming no intrinsic spin contributions at the classical level).

Physical interpretation

Think about what the diagonal terms actually tell you. Consider a region where only $E_x$ is nonzero. Then:

$T_{xx} = \frac{\epsilon_0}{2} E_x^2, \quad T_{yy} = T_{zz} = -\frac{\epsilon_0}{2} E_x^2$

The field pulls along its own direction (positive $T_{xx}$, a tension) and pushes perpendicular to itself (negative $T_{yy}$, $T_{zz}$, a pressure). Electric field lines behave like rubber bands under tension that also repel each other sideways. Magnetic field lines behave the same way. This tension-and-pressure picture is one of the most useful intuitions you can build for electromagnetic forces.

Derivation from Maxwell's equations

The stress tensor emerges from writing the Lorentz force density in a form that separates into a time derivative and a divergence.

1. Start with the Lorentz force density on charges and currents: $\vec{f} = \rho\vec{E} + \vec{J} \times \vec{B}$

2. Use Maxwell's equations to eliminate $\rho$ and $\vec{J}$ in favor of the fields:

   • $\rho = \epsilon_0 \nabla \cdot \vec{E}$
   • $\vec{J} = \frac{1}{\mu_0}\nabla \times \vec{B} - \epsilon_0 \frac{\partial \vec{E}}{\partial t}$

3. Substitute and rearrange. After considerable vector algebra (using product rules and the identity $\vec{E} \times (\nabla \times \vec{E})$, etc.), you arrive at: $\vec{f} = \nabla \cdot \overleftrightarrow{T} - \epsilon_0 \mu_0 \frac{\partial \vec{S}}{\partial t}$ where $\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}$ is the Poynting vector.

4. Recognizing the electromagnetic momentum density $\vec{g} = \mu_0 \epsilon_0 \vec{S} = \frac{1}{c^2}\vec{S}$, this becomes: $\vec{f} = \nabla \cdot \overleftrightarrow{T} - \frac{\partial \vec{g}}{\partial t}$

This is the conservation of electromagnetic momentum in local form. Every term has a clear meaning: the mechanical force density on matter equals the momentum flowing in through the stress tensor minus the rate at which field momentum is building up locally.

Conservation of momentum

Integrating over a volume $V$ bounded by surface $S$:

$\frac{d}{dt}(\vec{p}_{\text{mech}} + \vec{p}_{\text{field}}) = \oint_S \overleftrightarrow{T} \cdot d\vec{A}$

The left side is the total rate of change of mechanical plus field momentum inside $V$. The right side is the momentum flux flowing in through the boundary. In static or time-averaged situations (where $\partial \vec{g}/\partial t = 0$ or averages to zero), this simplifies to:

$\vec{F} = \oint_S \overleftrightarrow{T} \cdot d\vec{A}$

This is the key working formula. To find the force on any object, choose a closed surface around it, evaluate $\overleftrightarrow{T}$ on that surface (where you know the fields), and integrate.

Calculating force and torque

Force calculation steps:

1. Identify the object you want the force on.
2. Choose a closed surface $S$ surrounding the object. Pick a surface where the fields are easy to evaluate (often a sphere at large $r$, or a surface in a region of simple field geometry).
3. Compute $\vec{E}$ and $\vec{B}$ everywhere on $S$.
4. Build $T_{ij}$ at each point on the surface.
5. Evaluate $\vec{F} = \oint_S \overleftrightarrow{T} \cdot d\vec{A}$, where $d\vec{A} = \hat{n}\,dA$ points outward.
6. If the fields are time-varying, include the $-\frac{d}{dt}\int_V \vec{g}\,dV$ correction, or use time-averaged quantities for sinusoidal fields.

Torque about the origin is computed similarly:

$\vec{\tau} = \oint_S \vec{r} \times (\overleftrightarrow{T} \cdot d\vec{A})$

Relationship to energy density and momentum density

The stress tensor is the spatial part of a larger object. In the four-dimensional (relativistic) electromagnetic stress-energy tensor $T^{\mu\nu}$:

• $T^{00} = u = \frac{1}{2}\left(\epsilon_0 E^2 + \frac{1}{\mu_0}B^2\right)$ is the energy density.
• $T^{0i} = c\,g_i = \frac{S_i}{c}$ gives the momentum density (equivalently, energy flux divided by $c^2$).
• $T^{ij}$ are the spatial stress components defined above.

The trace of the spatial part is:

$T_{ii} = T_{11} + T_{22} + T_{33} = -\frac{1}{2}\left(\epsilon_0 E^2 + \frac{1}{\mu_0}B^2\right) = -u$

This result connects to the conformal invariance of classical electromagnetism: the full 4D trace $T^{\mu}{}_{\mu} = 0$ for the free electromagnetic field, reflecting the masslessness of the photon.

Conservation of energy

For completeness, the energy conservation (Poynting's theorem) sits alongside momentum conservation:

$\frac{\partial u}{\partial t} + \nabla \cdot \vec{S} = -\vec{J} \cdot \vec{E}$

The right side is the rate at which the field does work on free charges. Together, energy and momentum conservation form the divergence of the full stress-energy tensor:

$\partial_\mu T^{\mu\nu} = -f^{\nu}$

where $f^{\nu}$ is the four-force density on matter.

Stress tensor in linear media

In a linear, isotropic medium with permittivity $\epsilon$ and permeability $\mu$, the stress tensor generalizes to:

$T_{ij} = \epsilon E_i E_j + \frac{1}{\mu} B_i B_j - \frac{1}{2}\delta_{ij}\left(\epsilon E^2 + \frac{1}{\mu}B^2\right)$

This form applies inside homogeneous regions. At boundaries between different media, you need to be careful: the stress tensor is discontinuous across the interface because the material parameters jump. The force per unit area on the interface equals the difference in $\overleftrightarrow{T} \cdot \hat{n}$ evaluated on each side.

Stress tensor in anisotropic and nonlinear media

For anisotropic media, the constitutive relations involve tensors $\overleftrightarrow{\epsilon}$ and $\overleftrightarrow{\mu}$:

$\vec{D} = \overleftrightarrow{\epsilon} \cdot \vec{E}, \quad \vec{B} = \overleftrightarrow{\mu} \cdot \vec{H}$

The stress tensor must then be constructed from the full tensor products $E_i D_j$ and $H_i B_j$, and it is no longer guaranteed to be symmetric unless the permittivity and permeability tensors themselves are symmetric (which they are for lossless media).

For nonlinear media, where $\epsilon$ and $\mu$ depend on field strength, the stress tensor acquires additional terms related to electrostriction and magnetostriction. A proper treatment requires thermodynamic arguments (Helmholtz or Gibbs free energy formulations) to correctly account for the coupling between field energy and mechanical strain.

Applications in waveguides and cavities

In waveguides and resonant cavities, the stress tensor is useful for two main purposes:

• Radiation pressure on walls. The fields inside a waveguide or cavity exert forces on the conducting (or dielectric) boundaries. You can compute these by evaluating $\overleftrightarrow{T} \cdot \hat{n}$ just inside the boundary. For a perfect conductor, only the tangential $\vec{B}$ and normal $\vec{E}$ survive at the surface, which simplifies the calculation considerably.
• Mode analysis and energy flow. The time-averaged stress tensor $\langle T_{ij} \rangle$ gives the distribution of radiation pressure across the waveguide cross-section. This matters for understanding mechanical deformation of the guide under high power, and for optomechanical coupling in micro-cavities.

In cavity QED and optomechanical systems, the radiation pressure force derived from the stress tensor couples the electromagnetic cavity mode to the mechanical motion of a mirror or membrane. The force is proportional to the stored energy divided by the cavity length, and the stress tensor formalism provides the cleanest route to that result.

Applications in plasmas

Plasmas present a case where both the field stress tensor and the kinetic pressure of charged particles contribute to the total stress. The relevant equation of motion for a magnetized plasma element combines:

$\nabla \cdot \left(\overleftrightarrow{T}_{\text{EM}} + \overleftrightarrow{P}_{\text{kinetic}}\right) = \rho_m \frac{d\vec{v}}{dt}$

where $\rho_m$ is the mass density and $\overleftrightarrow{P}_{\text{kinetic}}$ is the kinetic pressure tensor.

The magnetic part of the stress tensor in a plasma is often decomposed into magnetic pressure ($B^2 / 2\mu_0$, acting isotropically perpendicular to $\vec{B}$) and magnetic tension ($B^2 / \mu_0$, pulling along field lines). This decomposition is central to magnetohydrodynamics (MHD) and explains phenomena like magnetic confinement in fusion devices, where the magnetic pressure balances the plasma kinetic pressure, and MHD instabilities, which arise when that balance is disrupted.