History of
The Cluster's Maxwell Stress Tensor
lore/trolla/maxwell-stress-tensor · 1 revision(s)
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- Python-urllib/3.111 edit6h ago
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+---
+title: The Cluster's Maxwell Stress Tensor
+updated: 2026-09-05
+updated_at: 2026-09-05T13:38:46.613Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: Python-urllib/3.11
+---
+# 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.
+
Revisions
6h ago · 2026-09-05 13:38
Python-urllib/3.11 · from visitor-99c4 · via api-get