# Maxwell stress tensor
The **Maxwell stress tensor** is a second-rank tensor, written \( T_{ij} \), that expresses the mechanical force per unit area transmitted by an electromagnetic field: contracting it with the normal of a surface gives the \( i \)-th component of the force across that surface, and integrating its flux over a closed surface gives the total electromagnetic force on the charges and matter inside.[1](https://farside.ph.utexas.edu/teaching/jk1/lectures/node11.html) • [2](https://ocw.mit.edu/courses/8-311-electromagnetic-theory-spring-2004/368a1711ad63a4f5a47d28201d00682e_ps5.pdf) It is the spatial part of the electromagnetic stress-energy tensor and appears in the local conservation law for electromagnetic momentum.
| Key fact | Detail |
|---|---|
| Definition (vacuum, SI) | \( T_{ij} = \varepsilon_0\left(E_iE_j - \tfrac{1}{2}\delta_{ij}E^2\right) + \tfrac{1}{\mu_0}\left(B_iB_j - \tfrac{1}{2}\delta_{ij}B^2\right) \)[1](https://farside.ph.utexas.edu/teaching/jk1/lectures/node11.html) |
| Force from a closed surface | \( F_i = \oint_S T_{ij}\,da_j \), with the area element pointing along the outward normal[2](https://ocw.mit.edu/courses/8-311-electromagnetic-theory-spring-2004/368a1711ad63a4f5a47d28201d00682e_ps5.pdf) |
| Momentum density it pairs with | \( \mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \mathbf{S}/c^2 \), where \( \mathbf{S} \) is the Poynting vector[3](https://www.feynmanlectures.caltech.edu/II_27.html) |
| Local force density | \( f_i = \partial_j T_{ij} - \varepsilon_0\mu_0\,\partial S_i/\partial t \)[4](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html) |
| Trace | \( T_{ii} = U \), the electromagnetic energy density[5](https://www2.physics.siu.edu/~shajesh/rp/files/EM-lnotes.pdf) |
| Covariant form | \( T^{\mu\nu} = \tfrac{1}{\mu_0}\left(F^{\mu\alpha}F_{\nu}^{\ \alpha} - \tfrac{1}{4}\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right) \), symmetric and trace-free[6](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/covariant-electromagnetism-cambridge-lectures.pdf) |
| Origin | Maxwell, Chapter V of Volume I of *A Treatise on Electricity and Magnetism*; in Art. 109 he says these considerations were central to Faraday's investigation[5](https://www2.physics.siu.edu/~shajesh/rp/files/EM-lnotes.pdf) |
## Definition and physical meaning
The tensor is quadratic in the field components. In vacuum and SI units its Cartesian components are
\[ T_{ij} = \varepsilon_0\left(E_iE_j - \tfrac{1}{2}\delta_{ij}E^2\right) + \frac{1}{\mu_0}\left(B_iB_j - \tfrac{1}{2}\delta_{ij}B^2\right), \]
where \( \delta_{ij} \) is the [Kronecker delta](https://www.edgepedia.org/kronecker-delta).[1](https://farside.ph.utexas.edu/teaching/jk1/lectures/node11.html) In Gaussian units the same object reads \( T_{\alpha\beta} = \tfrac{1}{4\pi}\left(E_\alpha E_\beta - \tfrac{1}{2}\delta_{\alpha\beta}E^2\right) \) for the electric part, and the matter version with \( D = \varepsilon E \), \( B = \mu H \) becomes \( T_{ij} = \tfrac{1}{8\pi}\delta_{ij}(\mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H}) - \tfrac{1}{4\pi}(E_iD_j + B_iH_j) \).[7](https://www.thphys.uni-heidelberg.de/~wegner/e03.dyn/El03Gese.pdf) • [2](https://ocw.mit.edu/courses/8-311-electromagnetic-theory-spring-2004/368a1711ad63a4f5a47d28201d00682e_ps5.pdf)
**Interpretation.** The component \( T_{\alpha\beta} \) gives the local flow rate of the \( \alpha \)-component of momentum density through a plane whose normal points in the \( \beta \)-direction; equivalently, \( T_{ij}\,da_j \) is the \( i \)-th component of force transmitted across an area element \( d\mathbf{a} \).[8](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_41_Poynting.html) • [4](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html) The diagonal components describe pressure or tension along field directions: magnetic field lines carry tension along their length and exert pressure across it, a quantitative version of Faraday's "tubes of force".[4](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html)
## Momentum conservation
Starting from the [Lorentz force](https://www.edgepedia.org/lorentz-force) density \( \rho\mathbf{E} + \mathbf{j}\times\mathbf{B} \) and eliminating the sources with [Maxwell's equations](https://www.edgepedia.org/maxwells-equations), the force density on matter can be written entirely in terms of the fields:[1](https://farside.ph.utexas.edu/teaching/jk1/lectures/node11.html) • [4](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html)
\[ f_i = \partial_j T_{ij} - \varepsilon_0\mu_0\,\frac{\partial S_i}{\partial t}, \qquad \mathbf{S} = \frac{\mathbf{E}\times\mathbf{B}}{\mu_0}. \]
Integrated over a volume \( V \), this is a local momentum-conservation law: the mechanical momentum of matter plus the field momentum, with density \( \mathbf{g} = \varepsilon_0\,\mathbf{E}\times\mathbf{B} = \mathbf{S}/c^2 \), changes only through the flux \( \oint_S T_{ij}\,da_j \) through the bounding surface.[3](https://www.feynmanlectures.caltech.edu/II_27.html) • [4](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html) The relation \( \mathbf{g} = \mathbf{S}/c^2 \) follows from the general theorem that any energy flow, divided by \( c^2 \), equals a momentum density; in electrodynamics the energy flux is the [Poynting vector](https://www.edgepedia.org/poynting-vector).[3](https://www.feynmanlectures.caltech.edu/II_27.html) In static situations the time-derivative term vanishes and the force on the enclosed charges reduces to the surface integral of the tensor alone.[2](https://ocw.mit.edu/courses/8-311-electromagnetic-theory-spring-2004/368a1711ad63a4f5a47d28201d00682e_ps5.pdf)
## Relation to the stress-energy tensor
The Maxwell stress tensor is the spatial block of the covariant electromagnetic stress-energy tensor
\[ T^{\mu\nu} = \frac{1}{\mu_0}\left(F^{\mu\alpha}F_{\nu}^{\ \alpha} - \frac{1}{4}\eta^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right), \]
which is symmetric and trace-free, \( T^{\mu}_{\ \mu} = 0 \).[6](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/covariant-electromagnetism-cambridge-lectures.pdf) Its \( 00 \) component is the energy density, its \( 0i \) components the momentum density and energy flux, and its \( ij \) components the Maxwell stress tensor; the covariant conservation law \( \partial_\mu T^{\mu\nu} = -F^{\nu}_{\ \mu}J^{\mu} \) combines energy and momentum conservation into one equation.[6](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/covariant-electromagnetism-cambridge-lectures.pdf) In three-dimensional terms, the trace of the stress tensor equals the energy density \( U = \tfrac{1}{2}(\mathbf{D}\cdot\mathbf{E} + \mathbf{B}\cdot\mathbf{H}) \).[5](https://www2.physics.siu.edu/~shajesh/rp/files/EM-lnotes.pdf)
## Worked results
**Capacitor plate.** For a parallel-plate capacitor with field \( E = \hat{z}\,\sigma/\varepsilon_0 \) between the plates and zero field elsewhere, the momentum density is zero and \( \hat{z}\cdot\mathbf{T} = -U\hat{z} \) inside the gap; the force per unit area on a plate comes out as \( F/A = U\hat{z} \), that is, the field energy density \( \sigma^2/2\varepsilon_0 \) acts as an outward pressure on each plate.[5](https://www2.physics.siu.edu/~shajesh/rp/files/EM-lnotes.pdf)
**Conductor surface.** For the electrostatic force on a plane piece of metal of area \( F \) in vacuum, evaluating the tensor flux gives \( K = \tfrac{1}{8\pi}E^2 n F \) in [Gaussian units](https://www.edgepedia.org/gaussian-units), where \( n \) is the surface normal; the pressure depends only on the normal field component at the surface.[7](https://www.thphys.uni-heidelberg.de/~wegner/e03.dyn/El03Gese.pdf)
**Radiation pressure.** For a plane wave incident on an absorbing screen, the stress-tensor component normal to the surface gives the radiation pressure, expressible in terms of the wave's energy density; the method extends to oblique incidence, reflecting surfaces, and spheres, and underlies calculations for solar sails, laser-trapped particles, and optical tweezers.[9](https://belashchenko.unl.edu/tutorial-maxwell-stress-tensor-and-radiation-pressure/) • [4](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html)
## Caveats and extensions
**Gauge freedom of the surface.** The interpretation of the tensor as mechanical stress transmitted by fields through empty space must be employed with understanding: one can add a constant to all components of the stress tensor without changing any calculated force density or total force, so only surface integrals over closed surfaces have unambiguous meaning.[10](https://ocw.mit.edu/courses/res-6-003-electromechanical-dynamics-spring-2009/dfbe1d68cb519b89503e93dfee2da213_chap08_sec_emd.pdf)
**Matter.** In a linear dielectric or magnetic medium the vacuum expression is replaced by forms involving \( \mathbf{D} \) and \( \mathbf{H} \), and in ponderable media several inequivalent stress tensors exist; a tensor derived from the generalized Lorentz law on polarization and magnetization differs from the well-known Abraham and Minkowski tensors, which have been at the center of a century-old controversy over field momentum in transparent materials.[2](https://ocw.mit.edu/courses/8-311-electromagnetic-theory-spring-2004/368a1711ad63a4f5a47d28201d00682e_ps5.pdf) • [11](https://arxiv.org/abs/1402.7261)
## References
1. [Electromagnetic Momentum Conservation, R. Fitzpatrick, University of Texas](https://farside.ph.utexas.edu/teaching/jk1/lectures/node11.html)
2. [8.311 Electromagnetic Theory, Problem Set 5, MIT OpenCourseWare](https://ocw.mit.edu/courses/8-311-electromagnetic-theory-spring-2004/368a1711ad63a4f5a47d28201d00682e_ps5.pdf)
3. [The Feynman Lectures on Physics, Vol. II, Ch. 27: Field Energy and Field Momentum](https://www.feynmanlectures.caltech.edu/II_27.html)
4. [Maxwell Stress Tensor and Field Momentum, physics2u](https://www.physics2u.com/vault/maxwell-stress-tensor-momentum.html)
5. [Notes on Electrodynamics, K. V. Shajesh, Southern Illinois University](https://www2.physics.siu.edu/~shajesh/rp/files/EM-lnotes.pdf)
6. [Electrodynamics lecture notes, A. Challinor, University of Cambridge](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/covariant-electromagnetism-cambridge-lectures.pdf)
7. [Electrostatics lecture notes, F. Wegner, Universität Heidelberg](https://www.thphys.uni-heidelberg.de/~wegner/e03.dyn/El03Gese.pdf)
8. [Poynting's Theorem and Maxwell's Stress Tensor, University of Virginia](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_41_Poynting.html)
9. [Tutorial: Maxwell Stress Tensor and Radiation Pressure, K. Belashchenko, University of Nebraska](https://belashchenko.unl.edu/tutorial-maxwell-stress-tensor-and-radiation-pressure/)
10. [Electromechanical Dynamics, Ch. 8, MIT OpenCourseWare](https://ocw.mit.edu/courses/res-6-003-electromechanical-dynamics-spring-2009/dfbe1d68cb519b89503e93dfee2da213_chap08_sec_emd.pdf)
11. [Electromagnetic Stress Tensor in Ponderable Media, arXiv:1402.7261](https://arxiv.org/abs/1402.7261)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electric and magnetic fields › Maxwell's equations and field formulations › Covariant formulation of electromagnetism › Electromagnetic stress–energy tensor*
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