Page 84 - Maxwell House

64 Chapter 2

2.3.2 Boundary Conditions on Normal Components of Field

Let we have an interface between two media with different parameters and small right cylinder
crossing the boundary between them as shown in Figure 2.3.2. Here is the side cylinder
3
surface of 2Δh height, ΔA is the cylinder base surfaces, and � is the unit normal to base.
rd
Applying 3 Maxwell’s equation from Table 1.7 to the cylinder internal fields we have
∯ ∘ = ∬ + ∬ + ∬ = (2.41)
1 2 3
11
Here is the free electrical charge inside the cylinder. The first and second integral term

in (2.41) is equal to
∬ ∘ � = ∆
1

� (2.42)
1
− ∬ ∘ � = − ∆
2

2
Here and is the normal component of vector D in medium 1 and 2, respectively. Since
2
1
→ 0 as ∆ℎ → 0 the limit of last integral term in (2.41) is equal to 0 as the cylinder height is
3
diminished. Therefore,
lim ∯ ∘ = ( − )∆ = (2.43)
∆ℎ→0 1 2
or

− 2 = lim (2.44)
1
∆→0 ∆
The right-hand limit can be non-zero if there are some free charges on the interface surface. We
denote the surface charge density by symbol
2
= lim [C/m ] (2.45)
∆→0 ∆
and can rewrite equation (2.44) as follows
− 2 = (2.46)
1

In term of the E-field

− = ⁄ (2.47)

2 2

1 1
0
It is clear that if the interface boundary is free of free charges
= (2.48)
1 1
2 2
The similar procedure applied to 4 Maxwell’s equation in Table 1.7 gives us
th
1 = (2.49)
2
1 = (2.50)
2
1
2
Equation (2.49) tells us that the normal component of magnetic inductance is always continuous
across the interface. We omitted free magnetic charges as a today fiction.
11 Recall that all bounded charges are included in displacement vector D.