This page talks briefly about tensors.
Flow of charge density is described through current density - a vector.
The flow of mememtum density is described through the Maxwell Stress Tensor.
Lets look at charge transport:
If the charge, of uniform density rho, flows uniformly in space and time, like wind with a velocity v. For a cross section A, the charge of the element is
Q = rho * v * dt * A
Current density is Charge/time/Area
J = Q/dt/A = rhp * v
This direction is parallel to v.
vector J = rho * vector v
To measure current density, we need to align the cross section to be perpendicular to the vector. Instead, it is also possible to get the current density by measuring the vector components in x, y, z directions.
Imagine J = (Jx, Jy, Jz)
Jx Ax = J Ax Cos(theta) = J.x Ax
We only measure the components of the vector in the unit vector direction and them combine them to get the full vector.
Now, look at parallel with momentum.
Assume that momentum flows uniformly in space and time. If the momentum was carried by matter ( like rain), the motion of the raindrops (vector v) would be parallel to the mementum being transported (vector p)
In the case of electromagnetic fields, the momentum might not be parallelto the direction the field is moving.
assume momentum density is vector g.
momentum vector p = vector g * v * dt * A
density of x momentum = x-momentum.dt.area = Px / dt / A = gx * v
This has a direction parallel to v and the magnitude as gx * v.
Same can be done for y and z components too. so the total momentum current density is vector g * vector v.
This is called outer product or tensor product.
James Clark Maxwell defined this with a negative sign, tensor T = -vector g * vector v.
$$ T = -\begin{pmatrix}
g_x \
g_y \
g_z
\end{pmatrix}
\begin{pmatrix}
v_x &
v_y &
v_z
\end{pmatrix} = -
\begin{pmatrix}
gx vx & gx vy & gx vz \
gy vx & gy vy & gy v_z \
gz vx & gz vy & gz vz
\end{pmatrix} $$
Now, imagine we work with force. The momentun is conserved in the absence of external force.
We can start with nothing, and have a momentum by applying force.
Force exerted by the changing momentum = F = p / dt = g * v .n * A = -T.n A
By newton’s third law, the external force = = T . n * A
If the surface is not uniform, we can integrate over the surface
F_total = integral_S T . n dA
Instead of integrating over volume, we need to itegrate the tensor over a surface.
An improvement is Griffith’s equation
$$ \vec Ttotal = on Surface \int \overleftrightarrow{T}\cdot \hat n dA - \epsilon \mu \frac{d}{dt}\int_v \vec S d^3 r $$
Griffith’s formula for T is below
$$ T = \epsilon ( \vec E \vec E - \frac{1}{2}E^2 I) + \frac{1}{\mu_0}(\vec B \vec B - \frac{1}{2}B^2 I) $$
Reference:
http://www2.oberlin.edu/physics/dstyer/Electrodynamics/MaxwellStressTensor.pdf