The Maxwell stress tensor describes how electromagnetic fields exert mechanical forces (pressures and tensions) on matter and space. It expresses the electromagnetic force density in terms of fields alone, enabling us to calculate forces on volumes via surface integrals of field quantities. This tensor is essential in astrophysics and plasma physics, where electromagnetic stresses govern dynamics such as jet collimation, magnetospheric forces, and radiation pressure.
We start from the volumetric Lorentz force density :
f ⃗ = ρ q E ⃗ + J ⃗ × B ⃗ . f = ρ q E + J × B .
Using Gauss's law and the Ampère-Maxwell law , we substitute charge density ρ q ρ q J ⃗ J
f ⃗ = ε 0 ( ∇ ⋅ E ⃗ ) E ⃗ + ( 1 μ 0 ∇ × B ⃗ − ε 0 ∂ E ⃗ ∂ t ) × B ⃗ f = ε 0 ( ∇ ⋅ E ) E + ( μ 0 1 ∇ × B − ε 0 ∂ t ∂ E ) × B
To simplify, consider the time derivative of the Poynting vector S ⃗ = 1 μ 0 E ⃗ × B ⃗ S = μ 0 1 E × B
∂ ∂ t ( E ⃗ × B ⃗ ) = ∂ E ⃗ ∂ t × B ⃗ + E ⃗ × ∂ B ⃗ ∂ t ∂ t ∂ ( E × B ) = ∂ t ∂ E × B + E × ∂ t ∂ B
Rearranged:
∂ E ⃗ ∂ t × B ⃗ = ∂ ∂ t ( E ⃗ × B ⃗ ) − E ⃗ × ∂ B ⃗ ∂ t ∂ t ∂ E × B = ∂ t ∂ ( E × B ) − E × ∂ t ∂ B
Using Faraday's law ∂ B ⃗ ∂ t = − ∇ × E ⃗ ∂ t ∂ B = − ∇ × E
∂ E ⃗ ∂ t × B ⃗ = ∂ ∂ t ( E ⃗ × B ⃗ ) + E ⃗ × ( ∇ × E ⃗ ) ∂ t ∂ E × B = ∂ t ∂ ( E × B ) + E × ( ∇ × E )
Substituting back into f ⃗ f
f ⃗ = ε 0 [ ( ∇ ⋅ E ⃗ ) E ⃗ − E ⃗ × ( ∇ × E ⃗ ) ] − 1 μ 0 B ⃗ × ( ∇ × B ⃗ ) − ε 0 ∂ ∂ t ( E ⃗ × B ⃗ ) f = ε 0 [ ( ∇ ⋅ E ) E − E × ( ∇ × E ) ] − μ 0 1 B × ( ∇ × B ) − ε 0 ∂ t ∂ ( E × B )
Just to make things look more symmetrical, we can throw in a term ( ∇ ⋅ B ⃗ ) B ⃗ ( ∇ ⋅ B ) B ∇ ⋅ B ⃗ = 0 ∇ ⋅ B = 0
f ⃗ = ε 0 [ ( ∇ ⋅ E ⃗ ) E ⃗ − E ⃗ × ( ∇ × E ⃗ ) ] + 1 μ 0 [ ( ∇ ⋅ B ⃗ ) B ⃗ − B ⃗ × ( ∇ × B ⃗ ) ] − ε 0 ∂ ∂ t ( E ⃗ × B ⃗ ) f = ε 0 [ ( ∇ ⋅ E ) E − E × ( ∇ × E ) ] + μ 0 1 [ ( ∇ ⋅ B ) B − B × ( ∇ × B ) ] − ε 0 ∂ t ∂ ( E × B )
Next, apply the vector identity:
∇ E 2 = 2 ( E ⃗ ⋅ ∇ ) E ⃗ + 2 E ⃗ × ( ∇ × E ⃗ ) , ∇ E 2 = 2 ( E ⋅ ∇ ) E + 2 E × ( ∇ × E ) ,
which when rearranged, gives:
E ⃗ × ( ∇ × E ⃗ ) = 1 2 ∇ E 2 − ( E ⃗ ⋅ ∇ ) E ⃗ , E × ( ∇ × E ) = 2 1 ∇ E 2 − ( E ⋅ ∇ ) E ,
as is similar for B ⃗ B
Substituting:
f ⃗ = ε 0 [ ( ∇ ⋅ E ⃗ ) E ⃗ + ( E ⃗ ⋅ ∇ ) E ⃗ ] + 1 μ 0 [ ( ∇ ⋅ B ⃗ ) B ⃗ + ( B ⃗ ⋅ ∇ ) B ⃗ ] − 1 2 ∇ ( ε 0 E 2 + 1 μ 0 B 2 ) − ε 0 ∂ ∂ t ( E ⃗ × B ⃗ ) f = ε 0 [ ( ∇ ⋅ E ) E + ( E ⋅ ∇ ) E ] + μ 0 1 [ ( ∇ ⋅ B ) B + ( B ⋅ ∇ ) B ] − 2 1 ∇ ( ε 0 E 2 + μ 0 1 B 2 ) − ε 0 ∂ t ∂ ( E × B )
Though this looks extremely complicated, it can actually be compactly written as
f ⃗ = ∇ ⋅ T − ε 0 μ 0 ∂ S ⃗ ∂ t , f = ∇ ⋅ T − ε 0 μ 0 ∂ t ∂ S ,
with the help of the Maxwell stress tensor T T
T i j = ε 0 ( E i E j − 1 2 δ i j E 2 ) + 1 μ 0 ( B i B j − 1 2 δ i j B 2 ) . T i j = ε 0 ( E i E j − 2 1 δ i j E 2 ) + μ 0 1 ( B i B j − 2 1 δ i j B 2 ) .
The total electromagnetic force on charges in volume V V S S
F ⃗ = ∫ V f ⃗ d τ = ∮ S T ⋅ d a ⃗ − ε 0 μ 0 d d t ∫ V S ⃗ d τ F = ∫ V f d τ = ∮ S T ⋅ d a − ε 0 μ 0 d t d ∫ V S d τ
where the divergence theorem converts the volume integral of ∇ ⋅ T ∇ ⋅ T S S
¶ Interpretation: Pressure and TensionPhysically, T i j T i j i i j j
The diagonal elements T i i T i i pressures (normal stresses).
The off-diagonal elements represent shear stresses (tensions).
Consider a uniform magnetic field B ⃗ = B 0 z ^ B = B 0 z ^ E ⃗ = 0 E = 0
T i j = − 1 2 μ 0 [ B 0 2 0 0 0 B 0 2 0 0 0 − B 0 2 ] T i j = − 2 μ 0 1 ⎣ ⎢ ⎡ B 0 2 0 0 0 B 0 2 0 0 0 − B 0 2 ⎦ ⎥ ⎤
This shows:
An outward pressure perpendicular to the field of magnitude B 0 2 / ( 2 μ 0 ) B 0 2 / ( 2 μ 0 )
A magnetic tension (negative pressure) along the field lines of the same magnitude.
What this shows us is that there is a magnetic pressure equal to the magnetic energy density pushing the field lines away from each other, while simultaneously keeping them straight through tension.
Radiation pressure near luminous objects can be analyzed using the Maxwell stress tensor.Magnetic tension and pressure are key to understanding solar flares, jets from AGN, and magnetospheric structures.The tensor formalism aids in modeling force balance in plasmas and the dynamics of magnetized astrophysical fluids .
D. J. Griffiths, Introduction to Electrodynamics , 4th ed.
J. D. Jackson, Classical Electrodynamics , 3rd ed.
F. S. Fitzpatrick, Maxwell’s Equations and the Principles of Electromagnetism