Derivation of the maxwell stress tensor for a linear dielectric. I make no guarantees about their legibility, or that they are totally free of errors. This chapter deals with conservation of energy, momentum. An introduction to tensors for students of physics and. Maxwells stress tensor is well known from electromagnetic theory. Their derivation is fairly opaque, at times even confusing, and their range of validity all but a. In the case of linear material equations the electromagnetic volume force equals zero. Maxwell stress tensor the maxwell stress tensor named after james clerk maxwell is a second rank tensor used in classical electromagnetism to represent the interaction between electromagnetic. The new equations include all maxwell solutions plus new ones, among which one may.
Chapter 6 maxwells equations in tensor form we now have learned the transformation properties of th electric and magnetic elds and together they form an antysymmetric tensor f f under poincare transformation. To check on this, recall for point charges we had ji ae av i a t 3r r at. The calculations of eigenvalues and reduced stresses in the case of tensor are shown in table 2. This is obtained from the basic contravariant field strength tensor by the substitutions. The eulerlagrange equation gets us back maxwell s equation with this choice of the lagrangian. Lecture notes on general relativity matthiasblau albert einstein center for fundamental physics. Osa electromagnetic stress tensor in ponderable media. A basic knowledge of vectors, matrices, and physics is assumed. A secondrank tensor whose product with a unit vector normal to a surface gives the force per unit area transmitted across the surface by an electromagnetic. The method is seldom discussed in introductory texts on. In analogy with the electrostatic case, the forces due to the magnetic field acting on the current distribution in a body can be obtained from a magnetic maxwell stress tensor, see j. It holds for all conceivable systems and is subject only to the validity of local equilibrium a constraint in frequency but not in the type and strength of. On deriving the maxwell stress tensor method for calculating.
The second method is less traditional but is easier to do. In this paper, we show the analysis of the influence of primary magnetic field orientation on the property of maxwell stress tensor for linear magnetoelasticity and static magnetoelasticity. Maxwell stress tensor, optical force, optical torque, divergence theorem some. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the. We derive an expression for the maxwell stress tensor in a magnetic dielectric medium specified by its permittivity. A semiintuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and similar higherorder vector products. Since the average of cos2 xover a single cycle is 1 2, the average momentum. The maxwell stress tensor has a great significance, for example for the calculation of forces in electromagnetic energy converters, and thus is standard content of advanced lectures during electrical engineering study. The canonical stress energy tensor from noethers theorem is not necessarily symmetric, and often needs to be improved with appropriate improvements terms. Electromagnetic stress tensor in ponderable media arxiv. However, the maxwell equations as they stand, written in the form given in equation 1.
In physics, the maxwell stress tensor is the stress tensor of an electromagnetic field. Further considerations concern maxwell stress tensors, in approximation to magnetostatic. The advantage of this strategy is that it is straightforward once the maxwell stress tensor is defined for the problem. Maxwells stress tensor and the forces in magnetic liquids. The method is seldom discussed in introductory texts on electromagnetism. Derivation of the maxwell stress tensor for a linear. The maxwell stress tensor named after james clerk maxwell is a mathematical object in physics, more precisely it is a second rank tensor used in classical electromagnetism to represent the interaction between electricmagnetic forces and mechanical momentum. Maxwells stress tensor article about maxwells stress. E has only an x component, and b only a y component all the offdiagonal i. But avoid asking for help, clarification, or responding to other answers. Thus, maxwell surface stress tensor, z component should be. This chapter deals with conservation of energy, momentum and. From maxwell stresses to nonlinear field equations arxiv. The electromagnetic stress tensor in the free space, known as maxwells stress tensor, is readily derived from the lorentz law of force in conjunction with the.
In this article we present a survey of the electromagnetic stress tensor and of the electromagnetic forces in. In classical electrostatics, electrostatic forces can be computed via the variational approach, for example, as in ref. The maxwell stress tensor is particularly useful for finding electromechanical boundary conditions in a concise form. Electromagnetic force and the maxwell stress tensor in. Covariant formulation of classical electromagnetism. Their derivation is fairly opaque, at times even confusing, and their range of validity all but a well kept secret. Maxwell stress tensor and electromagnetic waves 2 thus there is no momentum.
Canonical stress tensor for the free electromagnetic field. We work in the frame of an observer with 4velocity u. Construction of the stressenergy tensor we may build the stressenergy tensor by considering. Though extensively used in calculating optical force and torque acting on a material object illuminated by laser, the maxwell stress tensor mst method follows the electromagnetic linear and angular momentum balance that is usually derived in most textbooks for a continuous volume charge distribution in free space, if not resorting to the application of noethers theorem in electrodynamics. Maxwell s theory of electromagnetism into special relativity. The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism in particular, maxwells equations and the lorentz force in a form that is manifestly invariant under lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. The reader must be prepared to do some mathematics and to think. This force density can be expressed as the divergence of. The magnetic forces involve current densities which can be writt en as the fields as just described to.
Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Deriving maxwells equations from electromagnetic tensor. Maxwell stress dyadic in differentialform formalism. For a linear dielectric, the volumetric force density as obtained through the method of korteweighelmholtz and is given by. For the case of nonlinear materials such as magnetic iron with a bhcurve, the nonlinear maxwell stress tensor must be used. Physics in a gravitational field and tensor calculus 17 1 einstein equivalence principle. The korteweighelmholtz force density is obtained from a thermodynamic analysis of the dielectric. Introduction to tensor calculus for general relativity.
We find it useful to define the maxwell stress tensor as a function of the field quantities from which the force density can. Theory and applications of the maxwell stress tensor field precision. A modified maxwell stress tensor method for the evaluation of electromagnetic torque s. To recall, it is the limiting value of the ratio of force over area. Applications allow to evaluate the effectiveness of the method. We find it useful to define the maxwell stress tensor as a function of the field quantities from which the force density can be obtained by space differentiation. As we have previously emphasized, it is sufficient that the elastic portion of the stress is determined from a purely elastic strain energy function. But correct application of it to practical problems is by no means general knowledge even among experts. It follows at once that scalars are tensors of rank 0,0, vectors are tensors of rank 1,0 and oneforms are tensors of.
Just as before, the physical meaning is seen by integration over a volume. While the electromagnetic force is microscopically simply the lorentz force, its macroscopic form is more complicated, and given by expressions such as the maxwell stress tensor and the kelvin force. In undergraduate physics you learned that this was. Source free electrodyanmics contains but a single physical constant. The maxwell stress tensor may be used to calculate electric and magnetic forces on objects. Finally, we must write maxwells equations in covariant form.
The classical maxwell stress tensor or stressenergymomentum tensor is revisited by introducing the dyadic formalism to that of differential forms. Advanced texts often present the maxwell stress tensor as a mathematical abstraction without explaining why is is useful. The maxwell stress tensor given in terms of thermodynamic variables, the maxwell tensor is an unequivocally macroscopic, coarsegrained quantity. The divergence of maxwell stress tensor is the one which can acquire a physical identity and is electromagnetic body force density by an action at a distance like a gravity. In electromagnetism, the electromagnetic tensor or electromagnetic field tensor sometimes called the field strength tensor, faraday tensor or maxwell bivector is a mathematical object that describes the electromagnetic field in spacetime. I hope anybody will derive the most profit from comparing that to his own notes. Avoidance of contradictions in derivation and in usage of. Electromagnetic potentials making use of the homogeneous maxwell equations we want to introduce the electromagnetic potentials. A thermo dyna mic rederiv ation of the maxwell stress tensor is given in section i i i e, a n approach whic h is sim ilar in s pirit to that o f landau and lifshitz 2, 5, yet. And the force equation momentum conservation becomes.
Same state of stress is represented by a different set of components if axes are rotated. Of course when we want to calculate the total force on a body even ferromagnetic we do the surface integration of maxwell stress tensor in the air region, hence the material property is not involved. Mit opencourseware for any use or distribution of this. Maxwell stress tensor can be handled, inviting theorists and experimentalists alike to. Mechanical properties of the electromagnetic field densities. Pdf electromagnetic force and the maxwell stress tensor. Pdf on the physical meaning of maxwell stress tensor. The eulerlagrange equation gets us back maxwells equation with this choice of the lagrangian. It is important to emphasize that we have a lagrangian based, formal classical field theory for electricity and magnetism which has the four components of the 4vector potential as the independent fields. Thanks for contributing an answer to physics stack exchange. Electromagnetic stress tensor is called maxwell stress tensor. Lecture 7 maxwell stress tensor, electromagnetic momentum density and. The derivation proceeds from the generalized form of the lorentz law, which specifies the force exerted by the electromagnetic e and h fields on the polarization p and magnetization m of a ponderable medium.
Pdf while the electromagnetic force is microscopically simply the lorentz. Lecture 7 maxwell stress tensor, electromagnetic momentum density and angular momentum density, magnetic monopoles, diracs argument for the quantization of charge lecture 8 maxwells equations in potential form, gauge transformations, coulomb gauge, lorentz gauge. Dyadics, as originally introduced by gibbs to vector analysis, appear suitable companions to differential forms because of their coordinate free character. Jan 29, 2016 maxwell stress tensor the maxwell stress tensor named after james clerk maxwell is a second rank tensor used in classical electromagnetism to represent the interaction between electromagnetic. Maxwells theory of electromagnetism into special relativity. Here well look at a purely classical, nonrelativistic form of the tensor. Basic properties of dyadics together with some useful identities are first derived. The stress state is a second order tensor since it is a quantity associated with two directions two subscripts direction of the surface normal and direction of the stress. Maxwells stress tensor t let us use the lorentz force law to calculate the total electromagnetic force em fttot due to the totality of the electric charges contained within a source volumev. Various material models have been developed to describe rubber elastic materials. Stress balance principles 03 the cauchy stress tensor. The lectures are uploaded as pdf files, so you will need adobe acrobat reader in. Electrodynamicselectrostatic stress tensor wikibooks. The above derivation assumes complete knowledge of both.
If i understood correctly, the general case tensor which as you said contains e and d, b and h is called minkowsky stress tensor. The maxwell stress tensor for magnetoelastic materials. I dont understand what i am supposed to do to with this matrix to get the two maxwells equations below. Pdf electromagnetic force and the maxwell stress tensor in.
For example, we write the charge density which is part of the electrostatic force density rho times e in terms of the divergence of the efield. Im calculate the maxwell surface stress tensor of a particle over a surface in a pure electric filed. Maxwell stress tensor which is an elegant way of computing electromagnetic forces. Electrostatic forces in the poissonboltzmann systems. Maxwell stress tensor for a nonlinear media physics forums. The six independent elements of the symmetric maxwells stress tensor are. These expressions both make it simple to prove that the laws. Maxwell stress tensor for a monochromatic plane wave problem 9. Electromagnetic tensor in cylindrical coordinates from scratch. Construction of the stress energy tensor we may build the stress energy tensor by considering.
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