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.