E-StoreMagnetostatics 6
Boundary Condition for Magnetic Fields:
Similar to the boundary conditions in the electro static fields, here we will consider the behavior of 
and 
at the interface of two different media. In particular, we determine how the tangential and normal components of magnetic fields behave at the boundary of two regions having different permeabilities.
The figure 4.9 shows the interface between two media having permeabities 
and 
, 
being the normal vector from medium 2 to medium 1.
Figure 4.9: Interface between two magnetic media
To determine the condition for the normal component of the flux density vector , we consider a small pill box P with vanishingly small thickness h and having an elementary area 
for the faces. Over the pill box, we can write
....................................................(4.36)
Since h --> 0, we can neglect the flux through the sidewall of the pill box.
...........................(4.37) 
and 
..................(4.38) 
where 
and 
..........................(4.39)
Since is small, we can write 
or, 
...................................(4.40)
Since h -->0,
...................................(4.43)
We have shown in figure 4.8, a set of three unit vectors 
, 
and 
such that they satisfy 
(R.H. rule). Here is tangential to the interface and is the vector perpendicular to the surface enclosed by C at the interface.
The above equation can be written as 
or, 
...................................(4.44)
i.e., tangential component of magnetic field component is discontinuous across the interface where a free surface current exists.
If Js = 0, the tangential magnetic field is also continuous. If one of the medium is a perfect conductor Js exists on the surface of the perfect conductor.
In vector form we can write,
...................................(4.45)
Therefore, 
...................................(4.46)
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