# The Cluster's Maxwell Stress Tensor

A page about the Maxwell stress tensor — the tensor that describes the flow of momentum in electromagnetic fields.

## The Maxwell stress tensor

The Maxwell stress tensor is:
T_{ij} = epsilon_0 (E_i E_j - (1/2) delta_{ij} E^2) + (1 / mu_0) (B_i B_j - (1/2) delta_{ij} B^2)

It can be thought of as the electromagnetic momentum flux density. The divergence of T_{ij} gives the force density on charges:
f_i = nabla_j T_{ij} - epsilon_0 mu_0 (d/dt)(E x B)_i
In static situations: f_i = nabla_j T_{ij}.

In the cluster, the edit Maxwell stress tensor is an edit momentum flux density.

## The electromagnetic momentum

The electromagnetic momentum density is:
g = epsilon_0 (E x B) = S / c^2
where S = E x B / mu_0 is the Poynting vector. The total electromagnetic momentum is:
P_em = integral epsilon_0 (E x B) d^3 x

A charged capacitor has electromagnetic momentum in the space between its plates (the Feynman disk paradox illustrates this).

In the cluster, the edit electromagnetic momentum gives an edit momentum density.

## The force on a surface

The force on a surface S bounding a volume V is:
F_i = integral_S T_{ij} n_j dA
where n is the unit normal to the surface. This gives the radiation pressure on a surface.

For a perfect conductor with B normal to the surface and E tangent:
F = (1 / 2mu_0) B^2 (outward) = (1 / 2) epsilon_0 E^2 (outward)
The radiation pressure is P_rad = u = (1/2) epsilon_0 E_0^2 for a plane wave (absorbing surface). For a reflecting surface: P_rad = 2u.

In the cluster, the edit force on a surface gives an edit radiation pressure.

## Applications

- **Radiation pressure**: Solar sail propulsion, laser trapping of particles (optical tweezers)
- **MHD equilibrium**: The magnetic pressure B^2 / (2mu_0) and magnetic tension B^2 / mu_0 in plasma confinement
- **Capacitor force**: Attractive force between capacitor plates F = (1/2) Q^2 / (epsilon_0 A)
- **Waveguides**: The stress tensor determines the force on waveguide walls
- **General relativity**: The stress-energy tensor is a generalization of the Maxwell stress tensor

In the cluster, edit applications include:
- edit Radiation pressure
- edit MHD equilibrium
- edit Capacitor force
- edit Waveguides
- edit General relativity

## The Poynting vector relation

The momentum conservation equation is:
nabla . T - epsilon_0 mu_0 d/dt (E x B) = rho E + J x B = f_Lorentz
This is the statement that the rate of change of mechanical + electromagnetic momentum equals the force on charges:
d/dt (P_mech + P_em) = integral f_Lorentz d^3 x = integral_S T_{ij} n_j dA

In the cluster, the edit Poynting vector relation gives an edit conservation equation.

## This tensor

This page is about the Maxwell stress tensor. T_{ij} = epsilon_0 (E_i E_j - 1/2 delta_{ij} E^2) + 1/mu_0 (B_i B_j - 1/2 delta_{ij} B^2). Radiation pressure: P = u. The tensor is real.