# 7.5 Current Flow in a Good Conductor

When a dc voltage source is connected across the ends of a conducting wire, the current flowing through the wire is uniformly distributed over its cross section. That is, the current density J is the same along the axis of the wire and along its outer perimeter (Fig. 7-14(a)). This is not true in the ac case. As we see shortly, a time-varying current density is maximum along the perimeter of the wire and decreases exponentially as a function of distance toward the axis of the wire (Fig. 7-14(b)). In fact, at very high frequencies, most of the current flows in a thin layer near the wire surface, and if the wire material is a perfect conductor, the current flows entirely on the surface of the wire.

# Figure 7-14 Current density J in a conducting wire is (a) uniform across its cross section in the dc case, but (b) in the ac case, J is highest along the wire’s perimeter.

(a) dc case

(b) ac case

Before analyzing a wire with circular cross section, let us consider the simpler geometry of a semi-infinite conducting solid, as shown in Fig. 7-15(a). The solid’s planar interface with a perfect dielectric is the x−y plane. If at z=0− (just above the surface) an x-polarized electric field with E˜=x ˆ E 0 exists in the dielectric, a similarly polarized field is induced in the conducting medium and propagates as a plane wave along the +z direction. As a consequence of the boundary condition mandating continuity of the tangential component of E across the boundary between any two contiguous media, the electric field at z=0+ (just below the boundary) is E˜(0)=x ˆ E 0 also. The EM fields at any depth z in the conductor are given by

E˜(z)=x ˆ E 0 e−α z e−j β z,(7.83a)
H˜(z)=y ˆ E 0 η c e−α z e−j β z.(7.83b)

# Figure 7-15 Exponential decay of current density J˜x(z) with z in a solid conductor. The total current flowing through (a) a section of width w extending between z=0 and z=∞ is equivalent to (b) a constant current density J 0 flowing through a section of depth δ s.

(a) Exponentially decaying J˜x(z)

Figure 7 15-a Full Alternative Text

(b) Equivalent J0 over skin depth δs

Figure 7 15-b Full Alternative Text

From J=σ E, the current flows in the x direction, and its density is

J˜(z)=x ˆ J˜x(z),(7.84)

with

J˜x(z)=σ E 0 e−α z e−j β z=J 0 e−α z e−j β z,(7.85)

where J 0=σ E 0 is the amplitude of the current density at the surface. In terms of the skin depth δ s=1/α defined by Eq. (7.72) and using the fact that in a good conductor α=β as expressed by Eq. (7.77b), Eq. (7.85) can be written as

J˜x(z)=J 0 e−(1+j)z/δ s(A/m 2).(7.86)

The current flowing through a rectangular strip of width w along the y direction and extending between zero and ∞ in the z direction is

I˜=w∫0∞J˜x(z)d z=w∫0∞J 0 e−(1+j)z/δ s d z=J 0 w δ s(1+j)(A).(7.87)

The numerator of Eq. (7.87) is reminiscent of a uniform current density J 0 flowing through a thin surface of width w and depth δ s. Because J˜x(z) decreases exponentially with depth z, a conductor of finite thickness d can be considered electrically equivalent to one of infinite depth as long as d exceeds a few skin depths. Indeed, if d=3 δ s [instead of ∞ in the integral of Eq. (7.87)], the error incurred in using the result on the right-hand side of Eq. (7.87) is less than 5%; if d=5 δ s, the error is less than 1%.

The voltage across a length l at the surface (Fig. 7-15(b)) is given by

V˜=E 0 l=J 0 σ l.(7.88)

Hence, the impedance of a slab of width w, length l, and depth d=∞ (or, in practice, d>5 δ s) is

Z=V˜I˜=1+j σ δ s l w(Ω).(7.89)

It is customary to represent Z as

Z=Z s l w,(7.90)

where Z s is called the internal or surface impedance of the conductor and is defined as the impedance Z for a length l=1 m and a width w=1 m. Thus,

Since the reactive part of Z s is positive, Z s can be defined as

Z s=R s+j ω L s

with

R s=1 σ δ s=π f μ σ (Ω),(7.92a)
L s=1 ω σ δ s=1 2 μ π f σ (H),(7.92b)

where we used the relation δ s=1/α≈1/π f μ σ given by Eq. (7.77a). In terms of the surface resistance R s, the ac resistance of a slab of width w and length l is

R=R s l w=l σ δ s w(Ω).(7.93)

The expression for the ac resistance R is equivalent to the dc resistance of a plane conductor of length l and cross section A=δ s w.

The results obtained for the planar conductor can be extended to the coaxial cable shown in Fig. 7-16(a). If the conductors are made of copper with σ=5.8×10 7 S/m, the skin depth at 1 MHz is δ s=1/π f μ σ=0.066 mm, and since δ s varies as 1/f, it becomes smaller at higher frequencies. As long as the inner conductor’s radius a is greater than 5 δ s, or 0.33 mm at 1 MHz, its “depth” may be regarded as infinite. A similar criterion applies to the thickness of the outer conductor. To compute the resistance of the inner conductor, note that the current is concentrated near its outer surface and approximately equivalent to a uniform current flowing through a thin layer of depth δ s and circumference 2 π a. In other words, the inner conductor’s resistance is nearly the same as that of a planar conductor of depth δ s and width w=2 π a, as shown in Fig. 7-16(b). The corresponding resistance per unit length is obtained by setting w=2 π a and dividing by l in Eq. (7.93):

R 1′=R l=R s 2 π a(Ω/m).(7.94)

# Figure 7-16 The inner conductor of the coaxial cable in (a) is represented in (b) by a planar conductor of width 2 π a and depth δ s, as if its skin has been cut along its length on the bottom side and then unfurled into a planar geometry.

(a) Coaxial cable

(b) Equivalent inner conductor

Similarly, for the outer conductor, the current is concentrated within a thin layer of depth δ s on the inside surface of the conductor adjacent to the insulating medium between the two conductors, which is where the EM fields exist. The resistance per unit length for the outer conductor with radius b is

R 2′=R s 2 π b(Ω/m),(7.95)

and the coaxial cable’s total ac resistance per unit length is

This expression was used in Chapter 2 for characterizing the resistance per unit length of a coaxial transmission line.

Concept Question 7-6: How does β of a low-loss dielectric medium compare to that of a lossless medium?

Concept Question 7-7: In a good conductor, does the phase of H lead or lag that of E and by how much?

Concept Question 7-8: Attenuation means that a wave loses energy as it propagates in a lossy medium. What happens to the lost energy?

Concept Question 7-9: Is a conducting medium dispersive or dispersionless? Explain.

Concept Question 7-10: Compare the flow of current through a wire in the dc and ac cases. Compare the corresponding dc and ac resistances of the wire.