Ten years later, madariaga 1976 independently developed a similar method in elastodynamics. Its elements form a square array labelled by i and j that each refer to x, y, z directions and are defined in terms of the components of the electric field e and magnetic field b. The electromagnetic stressenergy tensor contains the negative of the classical maxwell stress tensor that governs the electromagnetic interactions. 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. Meep is a free and opensource software package for electromagnetics simulation via the finitedifference timedomain fdtd method features. The application gallery contains numerous simulation project files to get started quickly and easily.
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. Calculation of maxwell stress tensor using 3d fdtd for trapping. In this paper the model of trapping force on microsphere near focus in single optical tweezers is built by three dimensional finitedifference timedomain fdtd and maxwell stress tensor methods. The magnetic forces involve current densities which can be writt en as the fields as just described to. This chapter deals with conservation of energy, momentum and angular momentum in electromagnetic systems. They can be stated in integral form, in differential form a set of partial differential equations, and in tensor form. But avoid asking for help, clarification, or responding to other answers. Thanks for contributing an answer to physics stack exchange. The equations of mass and momentum conservation are derived, and the cauchy stress tensor makes its. The second method is less traditional but is easier to do. New forms of trapping force are proposed for the design of nearfield optical tweezers. Calculation of maxwell stress tensor using 3d fdtd. Finite difference time domain fdtd methods for solution.
In the absence of magnetization, the stress tensor given by eq. The maxwell stress tensor for magnetoelastic materials. Since the famous paper by yee 1966, the timedomain solution of maxwells equations has become very popular, and this initiated the finitedifference timedomain fdtd method taflove and hagness, 2000. Pdf electromagnetic force and the maxwell stress tensor. Methodology for optical force calculations lumerical support. Meep is a free and opensource software package for electromagnetics simulation via the finitedifference timedomain fdtd method spanning a broad range of applications. Calculation of maxwell stress tensor using 3d fdtd for trapping force in near field optical tweezers. The timeaveraged force acting on the center of mass of the particle is hfi z s htin ds 1 where sis a surface enclosing the particle, n is the unit vector perpendicular to the surface, and htiis the time. Although for magnetostatics, generally maxwell suggested. Torque through maxwell stress tensor comsol multiphysics. This page shows two techniques of calculating optical force on a particle. The maxwell stress tensor and the distributed lorentz force are applied to calculate.
We need only to calculate at most six components because the stress tensor is symmetric. Energy and momentum similar expressions are obtained for the magnetic term h. How are maxwells equations stated in the language of. Construction of the stressenergy tensorrst approach 2. 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. Determine all nine elements of the stress tensor in the region between the plates. Electromagnetic tensor in cylindrical coordinates from scratch. Stress measures this disambiguation page lists articles associated with the title stress tensor. The navierstokes equations are derived, along with their inertialess limit, the. The method is seldom discussed in introductory texts on electromagnetism.
In the maxwell stress tensor formulation, the total optical force acting on a par ticle due to a timeharmonic electromagnetic field is gi ven by. The maxwell stress tensor may be used to calculate electric and magnetic forces on objects. Code issues 95 pull requests 10 actions projects 0 security insights. In this section we use maxwells equations to derive the conservation law for lin ear momentum in an electromagnetic field. Fdtd numerical simulation of the trapping force of microsphere in. Just as before, the physical meaning is seen by integration over a volume. And the force equation momentum conservation becomes. E has only an x component, and b only a y component all the offdiagonal i. The symmetry of the tensor is as for a general stressenergy tensor in general relativity. The kx is a user forum for users who want to ask questions and exchange simulation tips with the larger lumerical community. Currents and the energymomentum tensor in classical field. Maxwell stress tensor which is an elegant way of computing electromagnetic forces. Computation of optical force on nanoparticles using.
Applications allow to evaluate the effectiveness of the method. Solution generally, the force on charges within a volume v is, f i s t da. Meep is a free and opensource software package for electromagnetics simulation via the finitedifference timedomain fdtd method spanning a broad range of applications key features. 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. Electromagnetic stress tensor is called maxwell stress tensor. In this paper a model of the trapping force on nanowires is built by three dimensional finitedifference timedomain fdtd and maxwell stress tensor methods, and the tightly focused laser beam is expressed by spherical vector wave functions vswfs. Im calculate the maxwell surface stress tensor of a particle over a surface in a pure electric filed. The trace of the energymomentum tensor is a lorentz scalar.
The calculations of eigenvalues and reduced stresses in the case of tensor are shown in table 2. Maxwells equations are the fundamental equations of classical electromagnetism and electrodynamics. Without the limitation of dipole approximation, the trapping force acting on a nanoparticle located in nearfield region can be solved by direct calculation of maxwell stress tensor using 3d fdtd method. The sign convention for the stress elements is that a positive force on a positive face or a negative force on a negative face is positive. Finite difference time domain fdtd methods for solution of. Maxwells equations properties of materials integral form equations in scalar form transverse mode maxwells equations in 2d fdtd methods divergencefree numerical stability 18th and 25th february, 2014 ucd p. How are maxwells equations stated in the language of tensors. I dont understand what i am supposed to do to with this matrix to get the two maxwells equations below. The maxwell tensor is also utilizable in calculation of total em force on a solid object, for example the force on electrically polarized body in external electric field or force on a magnet near metal body or another magnet. Electromagnetic stressenergy tensor, in relativistic physics. The net force can be found by integrating the mst over a closed surface surrounding the particle.
Meep is a free and opensource software package for electromagnetics simulation via the finitedifference timedomain fdtd method. In an ionic solution, the full stress tensor on the boundary should include an additional term accounting for the ionic pressure besides the conventional maxwell stress tensor. The maxwell stress tensor given in terms of thermodynamic variables, the maxwell tensor is an unequivocally macroscopic, coarsegrained quantity. The lumerical knowledge base contains the definitive reference guide on. The new forms are used to design nearfield optical trapping with a metalcoated fiber probe.
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. Calculation of maxwell stress tensor using 3d fdtd for. Finally, normal stress differencesare discussed, and the reader is warned about commonpitfalls in the mathematical modelingof complex. Advanced texts often present the maxwell stress tensor as a mathematical abstraction without explaining why is is useful. This chapter deals with conservation of energy, momentum. A modified maxwell stress tensor method for the evaluation of electromagnetic torque s. Thus, maxwell surface stress tensor, z component should be. The stressenergy tensor describes the flow of energy and momentum in spacetime. This chapter deals with conservation of energy, momentum and. Request pdf calculation of maxwell stress tensor using 3d fdtd for trapping force in nearfield optical tweezers new forms of trapping force are. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on.
Calculations of the trapping force of optical tweezers. Simulation in 1d, 2d, 3d, and cylindrical coordinates. Consider an infinite parallel plate capacitor with the lower plate at zd2 carrying the charge density \\sigma and the upper plate at zd2 carrying the charge density \\sigma. In relativistic physics, the electromagnetic stressenergy tensor is the contribution to the stressenergy tensor due to the electromagnetic field. Chapter maxwells equations and electromagnetic waves. Request pdf fdtd numerical simulation of the trapping force of microsphere in. The maxwell stress tensor named after james clerk maxwell is a symmetric secondorder tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. The finite integration technique as a general tool to. Same state of stress is represented by a different set of components if axes are rotated. 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. If an internal link led you here, you may wish to change the link to point directly to the intended article. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the. 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. Calculations of the trapping force of optical tweezers using. A derivation of the upperconvected maxwell model from the kinetic theory perspective is also provided. Electrodynamicselectrostatic stress tensor wikibooks. Mechanical properties of the electromagnetic field densities. In the case of linear material equations the electromagnetic volume force equals zero. Maxwell s equations properties of materials integral form equations in scalar form transverse mode maxwell s equations in 2d fdtd methods divergencefree numerical stability 18th and 25th february, 2014 ucd p. 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. 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. 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. The time averaged force f on a particle due to harmonic fields may be calculated from the maxwell stress tensor mst. Maxwell stress tensor for a monochromatic plane wave problem 9.
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. 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. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Osa electromagnetic stress tensor in ponderable media. Calculation of maxwell stress tensor using 3d fdtd for trapping force in nearfield optical tweezers article in materials science forum 697698. The components of the maxwell stress tensor as formulated in wikimedia. The maxwell tensor is also utilizable in calculation of total em force on a solid object, for example the force on electrically polarized body in external electric field. Electromagnetic field correlators, maxwell stress tensor. Fdtd simulation of trapping nanowires with linearly.
Deriving maxwells equations from electromagnetic tensor. Further considerations concern maxwell stress tensors, in approximation to magnetostatic. Computation of optical force on nanoparticles using locally. Calculation of the maxwell stress tensor and the poisson.
315 249 953 689 1463 1189 101 387 90 3 487 1459 1359 1131 491 119 1258 220 1245 27 1152 1192 1270 357 261 1279 527 222 561 183 1056 810 1493 971 1253 994 577 235 819 1377 238 314 1194 14 1143