E has only an x component, and b only a y component all the offdiagonal i. Electrostatics pdf electrostatics problem solving pdf. The second method is less traditional but is easier to do. Multiple part video deriving the maxwell stress tensor. Sep 07, 20 in classical electrostatics, electrostatic forces can be computed via the variational approach, for example, as in ref. Electrodynamicselectrostatic stress tensor wikibooks, open. Electrodynamicselectrostatic stress tensor wikibooks. Electrostatic forces in the poissonboltzmann systems. Torque through maxwell stress tensor comsol multiphysics.
The force vector can be obtained by surface integration of force density fi of equation 1. On the lagrangian electrostatics of elastic solids springerlink. Maxwells equations for electrostatics october 6, 2015 1 thedi. The total stress tensor, which includes the electrical effects, emerges in a natural way in this. I read previous posts on this forum on topic of maxwell stress tensor but still could not find what is wrong. The lecture notes were prepared in latex by james silva, an mit student, based upon handwritten notes. Introduction in almost all cases, mechanical systems with several degrees of freedom admit to a more. Pdf electromagnetic force and the maxwell stress tensor in. Detailed explanations of concepts, applications to numerous practical applications, problem sets, and a survey of dozens of application areas ranging from electronics to medicine. This is because they are written in the language of 3vectors. Meaning of the maxwell tensor we define the maxwell stress tensor as t ij. The maxwell stress tensor may be used to calculate electric and magnetic forces on objects. The maxwell stress tensor is particularly useful for finding electromechanical boundary conditions in a concise form.
This chapter deals with conservation of energy, momentum. Maxwell stress tensor can be handled, inviting theorists and experimentalists. Pdf download dep liquid actuation in viscous fluid in japanese, presented at annual meeting of the institute of electrostatics japan, iwate university, japan, september, 2004. Pdf download dispensing picoliter droplets using dielectrophoretic dep microactuation, microscale thermophysical engineering, vol. In relativistic physics, the electromagnetic stressenergy tensor is the contribution to the stressenergy tensor due to the electromagnetic field. Therefore, any surface integral involving the vector field. In classical electrostatics, electrostatic forces can be computed via the variational approach, for example, as in ref.
How does arise the force on a dielectric slab in a. It was shown that within the poissonboltzmann model two macroparticles with the same charge always repulsed each other in both isothermal and nonisothermal plasmas. 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. Keywords classical electromagnetism, maxwell equations, electrostatics, maxwell balance equation, maxwell stress tensor 1. By integrating the maxwell stress tensor over the surface of a dielectric particle, we derive the general expressions for dep force and rot torque generated by electrical fields of arbitrary configuration. 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.
The symmetry of the tensor is as for a general stressenergy tensor in general relativity. The maxwell stress tensor is a symmetric secondorder tensor used in. It turns out that when the quadrupole terms are kept in the multipole expansion of the laws of electrostatics, the solutions for the potential and the electric field are continuous functions at the surface. 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. 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. The results of the developed simulation procedures in openfoam are compared to the results of another simulation software.
Overview in this lecture, we will consider the spatial distribution of energy and momentum and their transport and conservation laws. Consider an inertial frame in which the 3velocity field of the particles is. Maxwell stress tensor in hydrodynamics iosr journals. We find it useful to define the maxwell stress tensor as a function of the field. Im trying to calculate the repelling force between two magnet rings with one levitated above the other.
Ma432 classical field theory trinity college, dublin. On the thermodynamic description of electrohydrodynamic. The latter describes the density and flux of energy and momentum in spacetime. 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. On the lagrangian electrostatics of elastic solids request pdf. Mit opencourseware for any use or distribution of this. Chapter 5 energy and momentum the equations established so far describe the behavior of electric and magnetic. On the thermodynamic description of electrohydrodynamic flows seyed reza mahmoudi, kazimierz adamiak, g. Various material models have been developed to describe rubber elastic materials.
As we have previously emphasized, it is sufficient that the elastic portion of the stress is determined from a purely elastic strain energy function. Maxwell s stress tensor rate of change of momentum due to em. The method is seldom discussed in introductory texts on electromagnetism. This tensor is related to the enstrophy in the same way, as magnetic stress tensor is associated to magnetic energy. Several examples are analyzed using the proposed approach.
In electrostatics and in magnetostatics, newtons 3rd law of motion always holds. Maxwell stress tensor for a monochromatic plane wave problem 9. We derive an expression for the maxwell stress tensor in a magnetic dielectric medium specified by its permittivity. This tensor identically vanishes out the body, whereas the maxwell stress. All the treatment is expounded in the reference configuration of the deformed body and the electromagnetic quantities and equations of interest are expressed in lagrangian form. Stressenergy tensor and conservation of energy and momentum christopher m. We explore the application of the maxwell stress tensor mst method to dielectrophoresis dep and electrorotation rot studies. The polarized interface between quadrupolar insulators. This tensor identically vanishes out the body, whereas the maxwell stress tensor is defined everywhere in the space. The stressenergy tensor describes the flow of energy and momentum in spacetime. Maxwell s equations em potentials equations of motion of particles in electromagnetic. The electromagnetic stressenergy tensor contains the negative of the classical maxwell stress tensor that governs the electromagnetic interactions. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the.
Pdf electromagnetic force and the maxwell stress tensor. However, the maxwell equations as they stand, written in the form given in equation 1. I am also to caculate the torque through maxwell stress tensor of the nps particles. This chapter deals with conservation of energy, momentum and angular momentum in electromagnetic systems.
Pdf while the electromagnetic force is microscopically simply the lorentz force. The field tensor was first used after the fourdimensional tensor formulation of special relativity was introduced by hermann minkowski. General expressions for dielectrophoretic force and. This force density and the stress tensor are part of a momentum balance equation derived from maxwell s equations. Here well look at a purely classical, nonrelativistic form of the tensor. 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. Under electroneutrality, the conductivity is proportional to this reduced concentration by. 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. They are a direct consequence of maxwells equations and the properties of matter.
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 advantage of this strategy is that it is straightforward once the maxwell stress tensor is defined for the problem. Pdf calculating maxwell stress tensor on the surface of a cell. The six independent elements of the symmetric maxwells stress tensor are. Jul 19, 2010 this problem has been bothering me for weeks. Despite the close similarity of the two tensors, they definitely differ from one another as they describe distinct physical phenomena. 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. Therefore, any surface integral involving the vector. Based on the authors course at the university of illinois.
This derivation requires to know vector calculus, linear algebra, and electrodynamics. May 22, 2016 multiple part video deriving the maxwell stress tensor. On the thermodynamic description of electrohydrodynamic flows. To the knowledge of the authors, the thermodynamics of the ehd. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. The electromagnetic energy tensor consider a continuous volume distribution of charged matter in the presence of an electromagnetic field. Note that the most general proper lorentz transformation can be written as a product of a 3rotation to align the new xaxis with the direction of motion, a boost along the new xdirection with velocity vand a second 3space rotation.
This force density and the stress tensor are part of a momentum balance equation derived from maxwells equations. On the lagrangian electrostatics of elastic solids. Electromagnetic force and the maxwell stress tensor in. To do this, the governing quasi electrostatic equation have been solved in consideration of cell shape. Thus analogous to the maxwell stress tensor associated to lorentz force ferraro and plumpton 1966 16 we get as the vorticity stress tensor.
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. Maxwell s theory of electromagnetism into special relativity. Let there be particles per unit proper volume unit volume determined in the local rest frame, each carrying a charge. The most widely used textbook on electrostatics available today. The equilibrium equations and the maxwell equations of electrostatics are derived here through a variational procedure. The trace of the energymomentum tensor is a lorentz scalar. Maxwell stress tensor can be handled, inviting theorists and experimentalists alike. I ask for your help to clear how to solve this problem.
Maxwell stress tensor, surface tension, and potential. Then, simple geometries with specific initial values and boundary conditions are calculated to test the new solvers. Introduction one of the simplest interactions of electromagnetic fields with matter is the in. Advanced texts often present the maxwell stress tensor as a mathematical abstraction without explaining why is is useful.
154 1044 1002 1158 986 997 618 135 273 1435 20 1473 1528 134 100 984 799 150 1459 385 661 1603 1155 1244 427 1381 654 849 971 1204 746 606 640