Force on a Charge Edit
When we want to discuss the force on a charge due to a charge distribution, there are two options. The first is a more traditional method: an integral over a volume containing the charge distribution. The second method is less traditional but is easier to do: a surface integral over a special stress tensor.
Volume Integral Version Edit
F
=
∫
V
ρ
E
d
V
{\displaystyle \mathbf {F} =\int _{V}\rho \mathbf {E} dV}
Electrostatic Stress Tensor Edit
T
E
=
1
4
π
[
E
x
2
−
E
2
2
E
x
E
y
E
x
E
z
E
x
E
y
E
y
2
−
E
2
2
E
y
E
z
E
x
E
z
E
y
E
z
E
z
2
−
E
2
2
]
{\displaystyle \mathbb {T} _{E}={\frac {1}{4\pi }}{\begin{bmatrix}E_{x}^{2}-{\frac {E^{2}}{2}}&E_{x}E_{y}&E_{x}E_{z}\\E_{x}E_{y}&E_{y}^{2}-{\frac {E^{2}}{2}}&E_{y}E_{z}\\E_{x}E_{z}&E_{y}E_{z}&E_{z}^{2}-{\frac {E^{2}}{2}}\end{bmatrix}}}
Surface Integral Version Edit
F
=
∫
S
T
n
d
A
{\displaystyle \mathbf {F} =\int _{S}\mathbb {T} \mathbf {n} dA}
The Maxwell Stress Tensor Edit
Tij is called the Maxwell Stress Tensor, it has two indices and is not a vector so is given a double arrow.
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
)
{\displaystyle \mathbb {T} _{ij}=\epsilon _{0}\left(E_{i}E_{j}-{\frac {1}{2}}\delta _{ij}E^{2}\right)+{\frac {1}{\mu _{0}}}\left(B_{i}B_{j}-{\frac {1}{2}}\delta _{ij}B^{2}\right)}
