11/14/2004 section 7_2 Maxwells Equations for Magnetostatics blank.doc 1/1 Jim Stiles The Univ. of Kansas Dept. of EECS 7-2 Maxwell’s Equations for Magnetostatics Reading Assignment: pp. 205-207 Recall that the static form of Maxwell’s Equations decoupled into Electrostatic and Magnetostatic equations. It’s now time to consider the Magnetostatic Equations! HO: Maxwell’s Equations for Magnetostatics Q: A: HO: The Integral Form of Magnetostatics11/14/2004 Maxwells equations for magnetostatics.doc 1/4 Jim Stiles The Univ. of Kansas Dept. of EECS Maxwell’s Equations for Magnetostatics From the point form of Maxwell’s equations, we find that the static case reduces to another (in addition to electrostatics) pair of coupled differential equations involving magnetic flux density ()rB and current density ()rJ: ()()()0r 0 x r rµ∇⋅= ∇ =BBJ Recall from the Lorentz force equation that the magnetic flux density ()rB will apply a force on current density ()rJ flowing in volume dv equal to: ()()()rx rdv=dF J B Current density ()rJ is of course expressed in units of Amps/meter2. The units of magnetic flux density ()rB are: 2Newton seconds WeberTeslaCoulomb meter meter⋅⋅11/14/2004 Maxwells equations for magnetostatics.doc 2/4 Jim Stiles The Univ. of Kansas Dept. of EECS * Recall the units for electric flux density ()rD are Colombs/m2. Compare this to the units for magnetic flux density—Webers/m2. * We can say therefore that the units of electric flux are Coulombs, whereas the units of magnetic flux are Webers. * The concept of magnetic flux is much more important and useful than the concept of electric flux, as there is no such thing as magnetic charge. We will talk much more later about the concept of magnetic flux! Now, let us consider specifically the two magnetostatic equations. * First, we note that they specify both the divergence and curl of magnetic flux density ()rB , thus completely specifying this vector field. * Second, it is apparent that the magnetic flux density ()rB is not conservative (i.e,()()0xr r 0µ∇= ≠BJ). * Finally, we note that the magnetic flux density is a solenoidal vector field (i.e., ()r0∇⋅ =B ).11/14/2004 Maxwells equations for magnetostatics.doc 3/4 Jim Stiles The Univ. of Kansas Dept. of EECS Consider the first of the magnetostatic equations: ()r0∇⋅=B This equation is sometimes referred to as Gauss’s Law for magnetics, for its obvious similarity to Gauss’s Law of electrostatics. This equation essentially states that the magnetic flux density does not diverge nor converge from any point. In other words, it states that there is no such thing as magnetic charge ! This of course is consistent with our understanding of solenoidal vector fields. The vector field will rotate about a point, but not diverge from it. Q: Just what does the magnetic flux density ()rB rotate around ? A: Look at the second magnetostatic equation!11/14/2004 Maxwells equations for magnetostatics.doc 4/4 Jim Stiles The Univ. of Kansas Dept. of EECS The second magnetostatic equation is referred to as Ampere’s Circuital Law: ()()0x r r Ampere's Lawµ∇=BJ This equation indicates that the magnetic flux density()rB rotates around current density ()rJ --the source of magnetic flux density is current!. ()rB()rJ11/14/2004 The Integral Form of Magnetostatics.doc 1/2 Jim Stiles The Univ. of Kansas Dept. of EECS The Integral Form of Magnetostatics Say we evaluate the surface integral of the point form of Ampere’s Law over some arbitrary surface S. ()()0xr rSSds dsµ∇⋅= ⋅∫∫ ∫∫BJ Using Stoke’s Theorem, we can write the left side of this equation as: ()()xr rSCds d∇⋅= ⋅∫∫ ∫BBAv We also recognize that the right side of the equation is: ()00rISdsµµ⋅=∫∫J where I is the current flowing through surface S. Therefore, combing these two results, we find the integral form of Ampere’s Law (Note the direction of I is defined by the right-hand rule): ()0rCdIµ⋅=∫B Av11/14/2004 The Integral Form of Magnetostatics.doc 2/2 Jim Stiles The Univ. of Kansas Dept. of EECS Amperes law states that the line integral of ()rB around a closed contour C is proportional to the total current I flowing through this closed contour (()rB is not conservative!). Likewise, we can take a volume integral over both sides of the magnetostatic equation ()r0∇⋅=B : ()r0Vdv∇⋅=∫∫∫B But wait! The left side can be rewritten using the Divergence Theorem: ()()rrVSdv ds∇⋅ = ⋅∫∫∫ ∫∫BBw where S is the closed surface that surrounds volume V. Therefore, we can write the integral form of ()r0∇⋅=B as: ()r0Sds⋅=∫∫Bw Summarizing, the integral form of the magnetostatic equations are: ()()0r 0 rSCds d Iµ⋅= ⋅=∫∫