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Example: Magnetostatic
Boundary Conditions
Consider two magnetic materials, separated by some boundary :
Throughout region 1, there is a constant magnetic field:
H 1 ( r ) = 3 ˆ α^ x + 5 ˆ α z ( z > 0 )
On the interface (i.e., boundary) between the two regions, there flows a surface current :
H 1 (^) ( r) = 3 α ˆ^ x + 5 α ˆ^ z A m/
H 2 ( r (^) )=??
J s (^) ( rb) = 2 a ˆ^ y A m/
z =
z
x
s y
z
r ˆ z A m
z
⎧^ >
J α
Q: What is H 2 ( r ) and B 2 ( r ) in region 2 ??
A: Let’s apply the boundary conditions and find out!
At the interface (i.e., z=0), we can state that:
H 2 (^) ( rb) = H 2 (^) ( z = (^0) ) = H 2 x( z = (^0) ) ˆ α x + H 2 y( z = (^0) ) ˆ α y + H 2 z( z = (^0) )ˆ α z
and:
B 2 (^) ( rb) = B 2 (^) ( z = (^0) ) = B 2 x ( z = (^0) ) ˆ α x + B 2 y( z = (^0) ) ˆ α y + B 2 z ( z = (^0) )ˆ α z
Therefore, we need to find the scalar components
H 2 x ( z = 0 ), B 2 x ( z = 0 ), etc.
First, we note thatˆ α z is normal to the interface, whileˆ α y and
ˆ α z are tangential.
Therefore:
H 2 x ( z = 0 ) = 1 and H 2 y( z = 0 )= 0
Q: But what about scalar component H 2 z( z = 0 )?
A: We can find it using our second boundary condition:
μ 1 H 1 (^) n ( rb) = μ 2 H 2 n ( rb)
From which we find:
( ) ( ) ( )
1 1 2 2 0 0 2
z z z z z z z
H z H z
H z
α α α α
And therefore:
( ) ( ) ( )
0 ˆ^ ˆ^0 2 ˆ^ ˆ
0 0 2 2
z z z z z z z
H z
H z
H z
α α α α
Thus, we find that:
2 (^0 )^2 (^0 )^2 (^0 )^2 (^0 )
(^103)
x x y y z z x z
z H z ˆ H z ˆ H z ˆ ˆ ˆ
= = = + = + = = +
H α α α α α
And since:
B 2 ( z = 0 ) = μ 2 H 2 ( z = 0 )
We find:
2 (^ )^0 (^ ) 0 0
x z x z
z ˆ ˆ
B α α α α
Q: But these are the values of the fields at the interface—
what are the fields throughout region 2?
A: Note that there are no conduction currents within region
- Thus, we find within region 2 :
∇ × H 2 ( r) = 0 ( z < 0 )
Note that a constant magnetic field will satisfy the above equation. Moreover, the following constant magnetic field will
likewise satisfy our boundary condition H 2 ( z = 0 ):
H 2 (^) ( r )= ˆ α x + 103 ˆ α z^ ⎡⎣A^ m⎤⎦
In other words, the value of the magnetic field at the boundary is likewise the value of the magnetic field
everywhere throughout region 2 ( H 2 ( r) is a constant vector
field!). The magnetic flux density is therefore:
B 2 ( r) = μ 2 H 2 ( r ) = 3 μ 0 ˆ α x + 10 μ 0 ˆ α z ⎡⎣W^ m^2 ⎤⎦
