Tutorial: Theoryandapplications …

1 Tutorial: Theory and applicationsof the Maxwell stress tensorStanley Humphries, Copyright 2012 Field PrecisionPO Box 13595, Albuquerque, NM 87192 : +1-505-220-3975 Fax: +1-617-752-9077E mail: Maxwell stress tensor may be used to calculate electric and magneticforces on objects. The method is seldom discussed in introductory texts onelectromagnetism. Advanced texts often present the Maxwellstress tensoras a mathematical abstraction without explaining why is is useful. In reality,the method is essential for practical force calculations innumerical codes. Inthis tutorial, I will review the theory and emphasize the specific reasons whythe method is used. To illustrate, I will concentrate on magnetostatic Lorentz force density is familiar from basic electromagnetism courses:f=j B.(1)The vector quantityfis the force per volume in a region with magnetic fluxdensityBand current densityj.

2 The total magnetic force on an object isgiven by a volume integral of the force density over its volume:F=Z Z ZdVj B.(2)It is important to recognize that the quantityjis the total current densitywithin the object, the sum of applied and atomic contributions. The appliedcurrent density arises from currents in coils. The magnitude, location anddirection of the applied current density follows from the known coil geometryand drive currents. Therefore, is relatively easy to apply Eq. 2 in numericalcodes to find coil forces. In contrast, volume integrals are not practical tofind force contributions from magnetically-active materials (field excluders,iron and permanent magnets). In this case, the current may beconcentratedin thin surface find an alternate force expression, we start from Ampere s law: 0j.(3)Note that the relative magnetic permeability ( r) does not appear in Eq.

3 3becausejrepresents the sum of all currents (applied and material). Substi-tution in Equation 2 gives an expression for the body force entirely in termsof the magnetic flux density:F=1 0Z Z ZdV( B) B.(4)Expanding the curl and the cross product, thexcomponent of Equation 4 isFx=1 oZ Z ZdV Bz Bx z Bz Bz x By By x+By Bx y!.(5)2 Consider taking the divergence of the following vector:Sx=1 0"x B2x B2y B2z2!+yBxBy+zBxBz#(6)The result is Sx=Bx Bxdx+ Bydy+ Bzdz! By By x Bz By x+By Bx y+Bz Bx z.(7)The condition B= 0 implies that the terms in parenthesis sum to that the remaining terms are identical to those in Equation 5. We cantherefore write thexcomponent of body force asFx=1 oZ Z ZdV Sx.(8)Finally, we can apply the divergence theorem to convert the volume integralto a surface integral:Fx=1 oZ ZdASx n.(9)The surface in Equation 9 may be any closed surface surrounding the quantitynis a unit vector pointing out of the surface.

4 Applying similaroperations to the other force components leads to the general force lawF=1 oZ ZdAS n.(10)The quantitySis the Maxwell stress tensor for magnetostatic fields:S=1 0 B2x B2/2 BxByBxBzByBxB2y B2/2 ByBzBzBxBzByB2z B2/2 ,(11)whereB2=B2x+B2y+ are three reasons why Equation 11 is better suited to a numericalcalculation than Equation 2: The accuracy of the integral is not affected by the distribution of cur-rent density within the onject. In particular, the thin surface layers ofmagnetically-active materials present no 1: Surface integral of the Maxwell stress )Valid surfaces forintegrals around magnetically-active : region surfaces boundedby air :surface of an air region that encloses the magnetic )Invalid surface for an integral. The enclosed material current isundefined on the common boundary. The surface need not coincide with the physical surface of the the object has sharp corners and regions of field enhancement, wecan improve accuracy be using a diagnostic surface removed from thephysical surface.

5 The integral depends only on the field distribution outside the object. Itis not necessary to know the exact current density distributions withincomplex anisotropic or nonlinear care must be exercised in applying the method. Because the integraldetermines the force on all applied and material currents inside the surface,we must avoid ambiguities in the enclosed current. The surface must be inan air region surrounding the object. Here, the term air implies that theregion contains no material currents ( = 0). Figure 1 illustrates the logicof the calculation. If we have an assembly of ferromagnetic and permanent-magnet objects surrounded by an air volume, then we can take the integralof Eq. 10 over any enclosing surface. One choice is the outer boundary ofthe assembly components (designated surfaceAin Fig. 1). In this case, weevaluate magnetic field values in the air elements near the surface to ensurethat the integral encloses all material currents of the object.

6 We could alsodefine an arbitrary surface by enclosing the assembly insidea diagnostic airregion. The integral over the surface markedBin Fig. 1 gives the same result(to within the numerical accuracy of field interpolations).Figure 1bshows acase where the stress tensor integral may not give the correct result. In thiscase the integral extends over the surface of ferromagneticregionAthat is in4contract with another iron or permanent-magnet region (B). The field valuesin regionBalong the common boundary include the effects of the surfacecurrents of both regions. The calculation gives the force onregionAplus anindeterminate portion of the force on