Page 24 - Fiber Optic Communications Fund

Electromagnetics and Optics 5

direction is shown in Fig. 1.4(b). From Eq. (1.13), we obtain

H ⋅ dL = H × circumference = I (1.14)
∮
L 1
or
I
H = . (1.15)
2r
Thus, the magnitude of the magnetic field intensity at a point is inversely proportional to its distance from the
conductor. Suppose the current is flowing in the z-direction. The z-component of the current density J may
z
be defined as the ratio of the incremental current ΔI passing through an elemental surface area ΔS =ΔXΔY
perpendicular to the direction of the current flow as the surface ΔS shrinks to zero,
ΔI
J = lim . (1.16)
z
ΔS→0 ΔS
The current density J is a vector with its direction given by the direction of the current. If J is not perpendicular
to the surface ΔS, we need to find the component J that is perpendicular to the surface by taking the dot
n
product
J = J ⋅ n, (1.17)
n
where n is a unit vector normal to the surface ΔS. By defining a vector ΔS =ΔSn,wehave

J ΔS = J ⋅ ΔS (1.18)
n
and the incremental current ΔI is given by
ΔI = J ⋅ ΔS. (1.19)

The total current flowing through a surface S is obtained by integrating,

I = J ⋅ dS. (1.20)
∫
S
Using Eq. (1.20) in Eq. (1.13), we obtain

H ⋅ dL = J ⋅ dS, (1.21)
∮ ∫
L1 S
where S is the surface whose perimeter is the closed path L .
1
In analogy with the definition of electric flux density, magnetic flux density is defined as

B = H, (1.22)

where is called the permeability. In free space, the permeability has a value
2
= 4 × 10 −7 N/A . (1.23)
0
In general, the permeability of a medium is written as a product of the permeability of free space and a
0
constant that depends on the medium. This constant is called the relative permeability ,
r
= . (1.24)
0 r