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
HB(r,) (r) 22
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 ()
ˆa µ n 1 H1t (r )
H2t (r )
H ()r 2n H (r ) = H (r ) + H ()r 2 2t 2n 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:
B ()r BH111(rr) = µ ( ) 1n
ˆan µ1 B (r ) 1t B2t (r ) B ()r 2n BH(rr) = µ ( ) 222
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
H (r ) 1tb µ J (r ) 1 sb
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