11/28/2004 Magnetic Boundary Conditions 1/6 Magnetic Boundary Conditions Consider the interface between two different materials with dissimilar permeabilities: HB11(r,) (r) µ1 HB22(r,) (r) µ2 Say that a magnetic field and a magnetic flux density is present in both regions. Q: How are the fields in dielectric region 1 (i.e., HB11()rr, ()) related to the fields in region 2 (i.e., HB22()rr, ())? A: They must satisfy the magnetic boundary conditions ! Jim Stiles The Univ. of Kansas Dept. of EECS 11/28/2004 Magnetic Boundary Conditions 2/6 First, let’s write the fields at the interface in terms of their normal (e.g.,Hn ()r ) and tangential (e.g.,Ht (r ) ) vector components: H r = H r + H r H1n ()r 1 ( ) 1t ( ) 1n () ˆan µ 1 H1t (r ) H2t (r ) H2n ()r H2 (r ) = H2t (r ) + H2n ()r µ 2 Our first boundary condition states that the tangential component of the magnetic field is continuous across a boundary. In other words: HH12tb(rr) = tb( ) where rb denotes to any point along the interface (e.g., material boundary). Jim Stiles The Univ. of Kansas Dept. of EECS 11/28/2004 Magnetic Boundary Conditions 3/6 The tangential component of the magnetic field on one side of the material boundary is equal to the tangential component on the other side ! We can likewise consider the magnetic flux densities on the material interface in terms of their normal and tangential components: BHrr= µ B1n ()r 111( ) ( ) ˆan µ 1 B1t (r ) B2t (r ) B2n ()r BH222(rr) = µ ( ) µ2 The second magnetic boundary condition states that the normal vector component of the magnetic flux density is continuous across the material boundary. In other words: BB12nb(rr) = nb( ) where rb denotes any point along the interface (i.e., the material boundary). Jim Stiles The Univ. of Kansas Dept. of EECS 11/28/2004 Magnetic Boundary Conditions 4/6 Since BH()rr= µ (), these boundary conditions can likewise be expressed as: HH12tb(rr) = tb( ) BB()rr() 12tb= tb µµ12 and as: BB12nb(rr) = nb( ) µµ11HHnb()rr= 22 nb() Note again the perfect analogy to the boundary conditions of electrostatics! Jim Stiles The Univ. of Kansas Dept. of EECS 11/28/2004 Magnetic Boundary Conditions 5/6 Finally, recall that if a layer of free charge were lying at a dielectric boundary, the boundary condition for electric flux density was modified such that: ˆ arnb⋅−⎣⎦⎡⎤DD12( ) ( r bsb) =ρ ( r) Dr12nb()−= Dr nb()ρ sb() r There is an analogous problem in magnetostatics, wherein a surface current is flowing at the interface of two magnetic materials: ˆan H1tb(r ) µ1 Jsb(r ) H2tb(r ) µ 2 In this case the tangential components of the magnetic field will not be continuous! Jim Stiles The Univ. of Kansas Dept. of EECS 11/28/2004 Magnetic Boundary Conditions 6/6 Instead, they are related by the boundary condition: ˆarnsx ()HH12( bb) −=( r) J( r b) This expression means that: 1) HH12tt()rrbb and ( ) point in the same direction. 2) HH12tt()rrbb and ( ) are orthogonal to Js ()rb . 3) The difference between HH12tt(rrbb) and () is Js (rb ) . Recall that H()r and Js (r ) have the same units— Amperes/meter! Note for this case, the boundary condition for the magnetic flux density remains unchanged, i.e.: BB12nb(rr) = nb( ) regardless of Js ()rb . Jim Stiles The Univ. of Kansas Dept. of EECS .