Maxwell Stress Theorem in Tensor Form

In classical electrodynamics the electromagnetic field carries not only energy but also momentum. The Lorentz force density

[
\mathbf{f}= \rho,\mathbf{E}+ \mathbf{J}\times\mathbf{B}
]

gives the mechanical force per unit volume acting on charges and currents. When the sources are distributed continuously, or when we are interested in the force that the field itself exerts on a piece of vacuum, evaluating the right‑hand side directly becomes cumbersome.

A more elegant route is to rewrite the force density entirely in terms of the fields (\mathbf{E}) and (\mathbf{B}). By substituting the charge and current densities with Maxwell’s equations

[
\nabla!\cdot!\mathbf{E}= \frac{\rho}{\varepsilon_{0}},\qquad
\nabla\times\mathbf{B}= \mu_{0}\mathbf{J}+ \mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t},
]

and applying a series of vector identities, the Lorentz force density can be expressed as the divergence of a second‑order tensor plus a term involving the time derivative of the Poynting vector. This tensor is the Maxwell stress tensor and it encapsulates the mechanical influence of the field in a compact, coordinate‑free form.

Definition of the Maxwell Stress Tensor

The Maxwell stress tensor (\mathbf{T}) is a symmetric (3\times3) tensor whose components are

[
T_{ij}= \varepsilon_{0}!\left(E_{i}E_{j}-\tfrac12\delta_{ij}E^{2}\right)
+\frac{1}{\mu_{0}}!\left(B_{i}B_{j}-\tfrac12\delta_{ij}B^{2}\right),
\qquad i,j\in{x,y,z}.
]

(E_{i}) and (B_{i}) are the Cartesian components of the electric and magnetic fields.
(E^{2}= \mathbf{E}!\cdot!\mathbf{E}) and (B^{2}= \mathbf{B}!\cdot!\mathbf{B}).
(\delta_{ij}) is the Kronecker delta, equal to 1 when (i=j) and 0 otherwise.

Because the tensor is symmetric ((T_{ij}=T_{ji})), it can be interpreted in the same way as the stress tensor of a fluid or an elastic solid:

Diagonal elements (T_{xx},T_{yy},T_{zz}) represent normal stresses—the pressure (or tension) that the field exerts along the corresponding axis.
Off‑diagonal elements (T_{ij};(i\neq j)) represent shear stresses, i.e. the tendency of the field to deform a material element sideways.

The Maxwell Stress Theorem

Combining the definition of (\mathbf{T}) with the Poynting vector

[
\mathbf{S}= \frac{1}{\mu_{0}},\mathbf{E}\times\mathbf{B},
]

one arrives at the differential form of the Maxwell stress theorem:

[
\boxed{;\mathbf{f}= \nabla!\cdot!\mathbf{T};-;\varepsilon_{0}\mu_{0},\frac{\partial\mathbf{S}}{\partial t};}
]

The first term, the divergence of (\mathbf{T}), is the field‑generated mechanical force density. The second term accounts for the rate of change of electromagnetic momentum density (\mathbf{g}= \varepsilon_{0}\mu_{0}\mathbf{S}).

Integrating over a fixed volume (V) and applying the divergence theorem gives the integral form:

[
\mathbf{F}{\text{total}}=
\oint{\partial V}!\mathbf{T}\cdot d\mathbf{a}
;-;\varepsilon_{0}\mu_{0},\frac{d}{dt}\int_{V}!\mathbf{S},dV .
]

The surface integral (\displaystyle\oint_{\partial V}\mathbf{T}\cdot d\mathbf{a}) is the net electromagnetic stress flux crossing the boundary (\partial V). Measuring the fields on the surface is sufficient to determine the total mechanical force acting on everything inside the volume.
The volume integral of (\mathbf{S}) is the total electromagnetic momentum stored in the region. Its time derivative subtracts the momentum that is being added to (or removed from) the field itself.

In a steady‑state situation where the fields do not vary with time, the second term vanishes and the total force reduces to a pure surface integral. This is the electromagnetic analogue of Newton’s third law: the mechanical force on matter equals the momentum flux carried by the field across the enclosing surface.

Physical Insight and Common Applications

1. Radiation Pressure on Mirrors

Consider a monochromatic plane wave of intensity (I) incident normally on a perfectly reflecting mirror. The fields on the incident side are (\mathbf{E}{\text{inc}},\mathbf{B}{\text{inc}}); on the reflected side they have the same magnitude but opposite direction for the Poynting vector. The relevant component of the stress tensor is

[
T_{zz}= \varepsilon_{0}!\left(E_{z}^{2}-\tfrac12E^{2}\right)
+\frac{1}{\mu_{0}}!\left(B_{z}^{2}-\tfrac12B^{2}\right).
]

Because the wave is transverse, (E_{z}=B_{z}=0) and the expression simplifies to

[
T_{zz}= -\frac{1}{2}!\left(\varepsilon_{0}E^{2}+ \frac{B^{2}}{\mu_{0}}\right)
= -\frac{I}{c},
]

where (c=1/\sqrt{\varepsilon_{0}\mu_{0}}). The pressure exerted on the mirror is the difference of (T_{zz}) just outside and just inside the surface, giving

[
p = \frac{2I}{c},
]

the familiar result that a perfectly reflecting surface experiences twice the momentum flux of the incident wave.

2. Force on a Small Charged Particle in a Non‑Uniform Field

When a charged particle is embedded in a rapidly varying electromagnetic field (e.g., in a plasma wakefield accelerator), it is often more convenient to treat the particle as the interior of a tiny Gaussian surface. The total force is then

[
\mathbf{F}= \oint_{\partial V}!\mathbf{T}\cdot d\mathbf{a},
]

where the surface integral can be evaluated using the known field values at the particle’s location. This bypasses the need to compute (\rho\mathbf{E}+\mathbf{J}\times\mathbf{B}) directly, which would require detailed knowledge of the particle’s charge distribution.

3. Electromagnetic Forces in Magnetohydrodynamics (MHD)

In conducting fluids the Lorentz force density (\mathbf{J}\times\mathbf{B}) appears in the momentum equation. Recasting it as (\nabla!\cdot!\mathbf{T}) makes it possible to treat magnetic stresses on the same footing as viscous stresses, facilitating the development of numerical schemes that conserve total momentum and energy.

Practical Tips for Using the Tensor Form

Choose a convenient surface: The integral form is most useful when the field is known on a simple geometric boundary (plane, sphere, cylinder). Symmetry often reduces the surface integral to a single component of (\mathbf{T}).
Check the time‑dependence: If the fields are time‑harmonic, the average over a full period eliminates the (\partial\mathbf{S}/\partial t) term, leaving only the time‑averaged stress tensor (\langle\mathbf{T}\rangle).
Mind the sign convention: The outward normal (d\mathbf{a}) points away from the volume. A positive flux of (\mathbf{T}) therefore corresponds to a force exerted on the volume by the surrounding field.
Units consistency: In SI units (\mathbf{T}) has units of pressure (N m(^{-2})). When working in Gaussian or Heaviside‑Lorentz units, the prefactors (\varepsilon_{0}) and (\mu_{0}) are replaced by the appropriate constants.

Summary

The Maxwell stress theorem in tensor form provides a powerful bridge between electromagnetic field theory and classical mechanics:

It replaces the Lorentz force density with the divergence of a symmetric second‑order tensor, making the force a purely field‑based quantity.
The integral version converts a volume problem into a surface problem, allowing forces to be computed from field values on a boundary alone.
By including the Poynting‑vector term, the theorem respects the conservation of total (mechanical + electromagnetic) momentum, even in time‑varying situations.

Because of these features, the Maxwell stress tensor is indispensable in areas ranging from optical manipulation and laser propulsion to plasma confinement and magnetohydrodynamic engineering. Mastery of its definition, interpretation, and application opens the door to a deeper, more unified understanding of how electromagnetic fields interact with matter.