# Maxwell Stress Tensor Calculator

T_ij = ε₀(E_iE_j − ½E²δ_ij) + (1/μ₀)(B_iB_j − ½B²δ_ij) from the full E and B vectors, with the 3×3 matrix, energy densities, trace check, and traction on a chosen surface.

> Canonical page: https://elysiatools.com/en/tools/maxwell-stress-tensor

- **Category:** Science & Education

- **Keywords:** maxwell stress tensor, electromagnetic stress, radiation pressure, field line tension, momentum flux, electromagnetism

## Overview

The Maxwell Stress Tensor Calculator computes the 3×3 electromagnetic stress tensor matrix, electric and magnetic energy densities, trace consistency check, and surface traction vector from three-dimensional electric and magnetic field components. By evaluating Maxwell stress expressions across configurable field units, it helps quantify electromagnetic momentum flux, radiation pressure, and field line tension in physical media and vacuum.

## Inputs

- **Ex** (number): e.g. 614
- **Ey** (number): 0 when absent
- **Ez** (number): 0 when absent
- **Bx** (number): 0 when absent
- **By** (number): e.g. 2.0474
- **Bz** (number): 0 when absent
- **E field unit** (select)
- **B field unit** (select)
- **Surface normal** (select)

## When to use

- Determining radiation pressure and momentum flux exerted by electromagnetic waves on planar boundaries.
- Evaluating electrostatic or magnetostatic forces and field line tension on charged or magnetized surfaces.
- Validating electromagnetic field calculations with energy density breakdowns and tensor trace checks.

## How it works

- Input the Cartesian vector components for both the electric field (Ex, Ey, Ez) and magnetic field (Bx, By, Bz), selecting appropriate measurement units.
- Select the surface normal orientation (+x, +y, or +z) to evaluate directional mechanical traction across a boundary.
- The tool computes the full 3×3 stress tensor matrix in N/m², electric and magnetic energy densities (u_E and u_B), the tensor trace check tr(T) = -(u_E + u_B), and the resulting surface traction vector.

## Use cases

- Calculating radiation pressure on solar sails and optical components under incident electromagnetic waves.
- Analyzing mechanical stress and electrostatic tension near high-voltage electrodes approaching dielectric breakdown.
- Verifying electrodynamics assignments and research models against analytical stress tensor formulas.

## Frequently asked questions

### What is the physical meaning of the Maxwell stress tensor?

It represents the electromagnetic momentum flux density and mechanical force per unit area exerted by electric and magnetic fields on matter.

### What does the tensor trace check verify?

In vacuum, the trace of the 3×3 Maxwell stress tensor equals the negative of the total electromagnetic energy density, confirming mathematical consistency.

### What does a negative diagonal value in the tensor indicate?

Negative diagonal elements represent isotropic or directional pressure compressing the surface, whereas positive diagonal elements represent tensile stress.

### Which field measurement units are supported?

Electric field inputs support V/m, kV/m, and MV/m, while magnetic field inputs support T, µT, and nT.

### How is the surface traction vector calculated?

Traction is obtained by multiplying the 3×3 stress tensor by the unit normal vector chosen for the target surface.

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