Maxwell's Stress Tensor
📂Electrodynamics Maxwell's Stress Tensor Definition The tensor T \mathbf{T} T Maxwell stress tensor .
T = T ↔ = ( T x x T x y T x z T y x T y y T y z T z x T z y T z z )
\mathbf{T}=\overleftrightarrow{\mathbf{T}}=\begin{pmatrix}
T_{xx} & T_{xy} & T_{xz}
\\ T_{yx} & T_{yy} & T_{yz}
\\ T_{zx} & T_{zy} & T_{zz}
\end{pmatrix}
T = T = T xx T y x T z x T x y T yy T zy T x z T yz T zz
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_{ij}=\epsilon_{0} \left( E_{i}E_{j}-\dfrac{1}{2}\delta_{ij}E^2 \right) + \dfrac{1}{\mu_{0}}\left(B_{i}B_{j}-\dfrac{1}{2}\delta_{ij}B^2 \right)
T ij = ϵ 0 ( E i E j − 2 1 δ ij E 2 ) + μ 0 1 ( B i B j − 2 1 δ ij B 2 )
Here, δ i j \delta_{ij} δ ij Kronecker delta .
Description 2 2 2 tensor is defined as above. It appears in the process of deriving the force experienced by a charge in a volume V \mathcal{V} V 2 2 2 A ⃗ \vec{A} A 2 2 2 A ↔ \overleftrightarrow{\mathbf{A}} A
The inclusion of the Kronecker delta in the formula makes the expressions for i = j i=j i = j i ≠ j i \ne j i = j
In the case of i = j i=j i = j
T x x = ϵ 0 ( E x 2 − 1 2 E 2 ) + 1 μ 0 ( B x 2 − 1 2 B 2 )
T_{xx}=\epsilon_{0} \left( {E_{x}}^2-\dfrac{1}{2}E^2\right) +\dfrac{1}{\mu_{0}}\left( {B_{x}}^2-\dfrac{1}{2}B^2 \right)
T xx = ϵ 0 ( E x 2 − 2 1 E 2 ) + μ 0 1 ( B x 2 − 2 1 B 2 )
since E 2 = E x 2 + E y 2 + E z 2 E^2={E_{x}}^2+{E_{y}}^2+{E_{z}}^2 E 2 = E x 2 + E y 2 + E z 2
T x x = ϵ 0 2 ( E x 2 − E y 2 − E z 2 ) + 1 2 μ 0 ( B x 2 − B y 2 − B z 2 )
T_{xx}=\dfrac{\epsilon_{0}}{2} \left( {E_{x}}^2-{E_{y}}^2-{E_{z}}^2\right) +\dfrac{1}{2\mu_{0}}\left( {B_{x}}^2-{B_{y}}^2 -{B_{z}}^2\right)
T xx = 2 ϵ 0 ( E x 2 − E y 2 − E z 2 ) + 2 μ 0 1 ( B x 2 − B y 2 − B z 2 )
In the case of i ≠ j i \ne j i = j
T x y = ϵ 0 ( E x E y ) + 1 μ 0 ( B x B y )
T_{xy}=\epsilon_{0}(E_{x}E_{y})+\dfrac{1}{\mu_{0}}(B_{x}B_{y})
T x y = ϵ 0 ( E x E y ) + μ 0 1 ( B x B y )
Inner Product The inner product of the Maxwell stress tensor T \mathbf{T} T a \mathbf{a} a 1 1 1
a ⋅ T = ( a x a y a z ) ( T x x T x y T x z T y x T y y T y z T z x T z y T z z ) = ( a x T x x + a y T y x + a z T z x , a x T x y + a y T y y + a z T z y , a x T x z + a y T y z + a z T z z )
\begin{align*}
\mathbf{a} \cdot \mathbf{T} &=\begin{pmatrix}
a_{x} & a_{y} & a_{z}
\end{pmatrix}\begin{pmatrix}
T_{xx} & T_{xy} & T_{xz}
\\ T_{yx} & T_{yy} & T_{yz}
\\ T_{zx} & T_{zy} & T_{zz}
\end{pmatrix}
\\ &= (a_{x}T_{xx}+a_{y}T_{yx}+a_{z}T_{zx},\ a_{x}T_{xy} + a_{y}T_{yy} + a_{z}T_{zy},\ a_{x}T_{xz} + a_{y}T_{yz} + a_{z}T_{zz})
\end{align*}
a ⋅ T = ( a x a y a z ) T xx T y x T z x T x y T yy T zy T x z T yz T zz = ( a x T xx + a y T y x + a z T z x , a x T x y + a y T yy + a z T zy , a x T x z + a y T yz + a z T zz )
( a ⋅ T ) j = ∑ i = x , y , z a i T i j
\left( \mathbf{a} \cdot \mathbf{T} \right)_{j}=\sum \limits_{i=x,y,z}a_{i}T_{ij}
( a ⋅ T ) j = i = x , y , z ∑ a i T ij
Divergence The divergence of T \mathbf{T} T j j j ∇ ⋅ T \nabla \cdot \mathbf{T} ∇ ⋅ T
( ∇ ⋅ T ) j = ϵ 0 [ ∇ i ( E i E j ) − 1 2 ∇ i δ i j E 2 ] + 1 μ 0 [ ∇ i ( B i B j ) − 1 2 ∇ i δ i j B 2 ] = ϵ 0 [ ∇ i E i E j + E i ∇ i E j − 1 2 ∇ j E 2 ] + 1 μ 0 [ ∇ i B i B j + B i ∇ i B j − 1 2 ∇ j B 2 ] = ϵ 0 [ ( ∇ ⋅ E ) E j + ( E ⋅ ∇ ) E j − 1 2 ∇ j E 2 ] + 1 μ 0 [ ( ∇ ⋅ B ) B j + ( B ⋅ ∇ ) B j − 1 2 ∇ j B 2 ]
\begin{align*}
& \left( \nabla \cdot \mathbf{T} \right)_{j} \\
=&\ \epsilon_{0} \left[ \nabla_{i}(E_{i}E_{j}) -\dfrac{1}{2}\nabla_{i}\delta_{ij}E^2 \right] + \dfrac{1}{\mu_{0}}\left[ \nabla_{i}(B_{i}B_{j})-\dfrac{1}{2}\nabla_{i}\delta_{ij}B^2 \right] \\
=&\ \epsilon_{0} \left[ \nabla_{i}E_{i}E_{j} +E_{i}\nabla_{i}E_{j} -\dfrac{1}{2}\nabla_{j}E^2 \right] + \dfrac{1}{\mu_{0}}\left[ \nabla_{i}B_{i}B_{j}+B_{i}\nabla_{i}B_{j}-\dfrac{1}{2}\nabla_{j}B^2 \right] \\
=&\ \epsilon_{0} \left[ (\nabla \cdot \mathbf{E})E_{j} +(\mathbf{E} \cdot \nabla) E_{j} -\dfrac{1}{2}\nabla_{j}E^2 \right] + \dfrac{1}{\mu_{0}}\left[ (\nabla \cdot \mathbf{B}) B_{j}+(\mathbf{B} \cdot \nabla) B_{j}-\dfrac{1}{2}\nabla_{j}B^2 \right]
\end{align*}
= = = ( ∇ ⋅ T ) j ϵ 0 [ ∇ i ( E i E j ) − 2 1 ∇ i δ ij E 2 ] + μ 0 1 [ ∇ i ( B i B j ) − 2 1 ∇ i δ ij B 2 ] ϵ 0 [ ∇ i E i E j + E i ∇ i E j − 2 1 ∇ j E 2 ] + μ 0 1 [ ∇ i B i B j + B i ∇ i B j − 2 1 ∇ j B 2 ] ϵ 0 [ ( ∇ ⋅ E ) E j + ( E ⋅ ∇ ) E j − 2 1 ∇ j E 2 ] + μ 0 1 [ ( ∇ ⋅ B ) B j + ( B ⋅ ∇ ) B j − 2 1 ∇ j B 2 ]
