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Maxwell stress tensor

Posted on Edited on

物理图像

想像在给定空间内的连续介质，未知处的电荷密度是ρ。由于电荷的存在，电场和磁场将对此处微元产生力的作用。可以证明，这个电磁场力等效为一个作用于此微元表面的应力张量

推导 (主要参考David J. Griffiths的《introduction to electrodynamics》，恰好Wikipedia上也是此书一样的推导，原文抄在下面)

Maxwell stress tensor

The Maxwell stress tensor (named after James Clerk
Maxwell) is a
symmetric second-order tensor used in
classical
electromagnetism
to represent the interaction between electromagnetic forces and
mechanical momentum. In simple situations,
such as a point charge moving freely in a homogeneous magnetic field, it
is easy to calculate the forces on the charge from the Lorentz force
law{}. When
the situation becomes more complicated, this ordinary procedure can
become impossibly difficult, with equations spanning multiple lines. It
is therefore convenient to collect many of these terms in the Maxwell
stress tensor, and to use tensor arithmetic to find the answer to the
problem at hand.

In the relativistic formulation of electromagnetism, the Maxwell's
tensor appears as a part of the electromagnetic stress–energy
tensor
which is the electromagnetic component of the total stress–energy
tensor. The
latter describes the density and flux of energy and momentum in
spacetime.

Contents

Motivation

Lorentz force (per unit 3-volume) f on a continuous charge
distribution{}
(charge density ρ) in motion.
The 3-current density J
corresponds to the motion of the charge element dq in volume
element dV and varies
throughout the continuum.

As outlined below, the electromagnetic force is written in terms of
E and B. Using vector
calculus and Maxwell's
equations, symmetry
is sought for in the terms containing E and B, and introducing
the Maxwell stress tensor simplifies the result.

Maxwell's equations in SI units in vacuum\ (for reference)

Name Differential form
Gauss's law (in vacuum)
Gauss's law for magnetism
Maxwell–Faraday equation (Faraday's law of induction)
Ampère's circuital law (in vacuum) (with Maxwell's correction)

  1. Starting with the Lorentz
    force     law

    the force per unit volume is

  2. Next, ρ and J can be replaced by the fields E and B,
    using Gauss's law and
    Ampère's circuital
    law    :

  3. The time derivative can be rewritten to something that can be
    interpreted physically, namely the Poynting
    vector    . Using the product
    rule     and Faraday's law of
    induction
    gives

    and we can now rewrite f as

    then collecting terms with E and B gives

  4. A term seems to be "missing" from the symmetry in E and B, which can be achieved by inserting [(∇ ⋅ B)B] because of Gauss' law for magnetism:

    Eliminating the curls (which are fairly complicated to calculate), using the vector calculus identity{}

    leads to:

  5. This expression contains every aspect of electromagnetism and
    momentum and is relatively easy to compute. It can be written more
    compactly by introducing the Maxwell stress tensor,

    All but the last term of f can be written as the tensor
    divergence of the Maxwell stress
    tensor, giving:

    As in the Poynting's theorem, the second term on the right side of the above equation can be interpreted as the time derivative of the EM field's momentum density, while the first term is the time derivative of the momentum density for the massive particles. In this way, the above equation will be the law of conservation of momentum in classical electrodynamics.

    where the Poynting vector has been introduced

in the above relation for conservation of momentum, is the momentum flux density and plays a role similar to
in Poynting's theorem.

The above derivation assumes complete knowledge of both ρ and J (both free and bounded charges and currents). For the case of nonlinear materials (such as magnetic iron with a BH-curve), the nonlinear Maxwell stress tensor must be used.

Equation

In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. As derived above in SI units, it is given by:

where ε0 is the electric constant{} and μ0 is the magnetic constant{}, E is the electric field, B is the magnetic field and δij is Kronecker's delta{}. In Gaussian cgs unit, it is given by:

where H is the magnetizing field{}.

An alternative way of expressing this tensor is:

where ⊗ is the dyadic product{}, and the last tensor is the unit dyad:

The element ij of the Maxwell stress tensor has units of momentum per unit of area per unit time and gives the flux of momentum parallel to the ith axis crossing a surface normal to the jth axis (in the negative direction) per unit of time.

These units can also be seen as units of force per unit of area (negative pressure), and the ij element of the tensor can also be interpreted as the force parallel to the ith axis suffered by a surface normal to the jth axis per unit of area. Indeed, the diagonal elements give the tension (pulling) acting on a differential area element normal to the corresponding axis. Unlike forces due to the pressure of an ideal gas, an area element in the electromagnetic field also feels a force in a direction that is not normal to the element. This shear is given by the off-diagonal elements of the stress tensor.

Magnetism only

If the field is only magnetic (which is largely true in motors, for instance), some of the terms drop out, and the equation in SI units becomes:

For cylindrical objects, such as the rotor of a motor, this is further simplified to:

where r is the shear in the radial (outward from the cylinder) direction, and t is the shear in the tangential (around the cylinder) direction. It is the tangential force which spins the motor. Br is the flux density in the radial direction, and Bt is the flux density in the tangential direction.

In electrostatics

In electrostatics the effects of magnetism are not present. In this case the magnetic field vanishes,
, and we obtain the electrostatic Maxwell stress tensor. It is given in component form by

and in symbolic form by

where is the appropriate identity tensor (usually ).

Eigenvalue

The eigenvalues of the Maxwell stress tensor are given by:^[*[citation\ needed]{title=”This claim needs references to reliable sources. (July 2015)”}*]^

Missing or unrecognized delimiter for \left{\lambda} = \left{ {- \left( {\frac{\epsilon_{0}}{2}E^{2} + \frac{1}{2\mu_{0}}B^{2}} \right),\ \pm \sqrt{\left( {\frac{\epsilon_{0}}{2}E^{2} - \frac{1}{2\mu_{0}}B^{2}} \right)^{2} + \frac{\epsilon_{0}}{\mu_{0}}\left( {\mathbf{E} \cdot \mathbf{B}} \right)^{2}}} \right}

These eigenvalues are obtained by iteratively applying the Matrix Determinant Lemma, in
conjunction with the Sherman-Morrison Formula{}.

Noting that the characteristic equation matrix, ,
can be written as

where

we set

Applying the Matrix Determinant Lemma once, this gives us

Applying it again yields,

From the last multiplicand on the RHS, we immediately see that is one of the eigenvalues.

To find the inverse of , we use the Sherman-Morrison formula:

Factoring out a term in the determinant, we are left with finding the zeros of the rational function:

Thus, once we solve

we obtain the other two eigenvalues.

References

1 David J. Griffiths, "Introduction to Electrodynamics" pp. 351–352, Benjamin Cummings Inc., 2008

2 John David Jackson, "Classical Electrodynamics, 3rd Ed.", John Wiley & Sons, Inc., 1999.

3 Richard Becker, "Electromagnetic Fields and Interactions", Dover Publications Inc., 1964.