An AC current I of frequency ω flows in a wire of radius a and conductivity σ (considered to be real and independent of frequency in the range considered). Find the impedance per unit length of the wire as a function of the frequency.
1

The basic issue with estimating the effective impedance of the wire actually depends on the choice of the boundary conditions. In fact, since induction effects lead to an electric field which, for ω = 0, is no longer uniform but depends on the radius, i.e.
E z
= E z
( r, t ) being z the coordinate along the wire. Thus, the voltage drop per unit length v = ∆ V / ∆ z is not unambiguously defined anymore, and the impedance per unit length Z can not be simply defined as Z = v/I , with I the current intensity given by
I = I ( t ) =
Z a
σE z
( r, t )2 πrdr .
0
(1)
There are two possible approaches which might (or might not) correspond to real situations. The first approach is to measure the voltage drop at the surface of the wire , i.e. at r = a . Performing such measurement as the frequency ω is varied implies that the field at the wire surface E z
( a, t ) is controlled and measured (that is to say that the measurements are made by keeping fixed the field just outside the wire). In this case, the boundary condition we fix is E z
( a ) = E
0
The second approach, following Jackson and Z = E
0
/I .
1 , is to define the impedance Z = R − i X from the total
(space-integrated) average values of the dissipated power and of the energy density inside the system.
The real part, i.e. the resistance, is thus defined as
R = Re Z =
1 h I 2 i
Z a
σ E 2 z
( r, t ) 2 πrdr =
0
1
| I
˜
| 2
Z a
σ |
˜ z
( r ) | 2 2 πrdr ,
0
(2) where the last expression holds if the oscillating current and fields are written as I = I ( t ) =
Re( ˜
− iωt ), E z
= E z
( r, t ) = Re( ˜ z
( r ) e
− iωt ). In this approach, we thus assume that the wire is connected to an ideal current generator. The imaginary part of the impedance, i.e. the “reactance”, is defined in terms of the electric and magnetic energy of the system:
X = Im Z =
2 ω h I 2 i
Z a
( h u m i − h u e i )2 πrdr ,
0
(3) that corresponds to the definitions of inductance and capacitance, following from − i X ≡ − iω L −
1 / ( iω C ) as
L =
2 h I 2 i
Z a
0
B
2
2 µ
0
2 πrdr ,
The basic equations for our system are
C =
2 ω 2 h I 2 i
Z a
0 ǫ
0 h E
2 i
2 πrdr .
2
(4)
∇ × E = − ∂ t
B , ∇ × B = µ
0
J = µ
0
σ E , (5) the displacement current being negligible in a good conductor. Then, assuming ∇ · E = 0 we get a diffusion equation for E :
∇ 2
E = µ
0
σ∂ t
E ,
1 J. D. Jackson, Classical Electrodynamics , 3rd Ed., par.6.9.
(6)
2

which for oscillating fields becomes
∇ 2 ˜
= − iωµ
0
σ
˜
≡ α 2 ˜
, α =
1 − i
,
δ
(7) where we introduced the (collisional) skin depth
δ =
2
µ
0
σω
1 / 2
.
(8)
Assuming cylindrical geometry (i.e.
E = E z
( r, t )), we eventually obtain
∇ 2 E z
=
1 r
∂ r
( r∂ r
E z
) = α 2 E z
.
(9)
Although this equation has an exact solution in terms of special functions (i.e. Bessel functions of complex argument), we proceed with two approximate methods correspondingly to the low-frequency
(LF) and high-frequency (HF) limits, respectively.
In the LF limit, we use the slowly varying current approximation. We assume that to lowest order the field is uniform, i.e.
E
0 z
= Re ( E
0 e
− iωt ). The associated current density J
0 z
= σE
0 z
= σE
0 generates a solenoidal magnetic field field E
1 z such that − ∂ r
E
1 z
= − ∂ t
B
1 φ
B
1 φ
, i.e.
= µ
0
− ∂ r rJ
0 z
E
1 z
/ 2, which in turns leads to an inductive electric
= + iω
˜
1 φ
. This gives as the first correcting term for the electric field
E
1 z
( r ) = − i ( r 2 / 2 δ 2 ) E
0
E
1 z
(0) .
(10)
E
1 z
(0) at the origin implies that either the first or the second choice for the boundary conditions are made. In the first case (“voltage” approach), we
E z
( a ) = ˜
0 z
( a ) + ˜
1 z
( a ) = E
0
E
1 z
( a ) = 0, thus to order ∼ a 2 /δ 2 the electric field is given by
E
0
E
1 z
= E
0
1 − i r 2 −
2 δ 2 a 2
.
(11)
To the same order of approximation, the impedance is given by Z = E
0
/I where the current I is calculated as the flow of σ ( E
0
+ E
1
) through the wire. Thus
Z
−
1 = σ
Z a
0
1 − i r 2 −
2 δ 2 a 2
2 πrdr = πa 2 σ 1 + ia
4 δ
2
2
.
(12)
By noticing that R
0
= ( πa 2 σ )
−
1 is the static (DC) resistance per unit length, we obtain
Z ≃ R
0
1 − ia 2
4 δ 2
≡ R
0
− iω L , L =
µ
0
8 π
.
(13)
We may thus interpret the effect of magnetic induction as the appearance of a parasitic inductance
L , in addition to the DC resistance.
3

A parasitic resistance appears to the next order in the series expansion. To this aim, we calculate the next order terms for the magnetic and the electric field,
∇
∇
×
×
B
E
2
2
= µ
= −
0
∂
σ E t
B
1
2
,
,
B
2 φ
E
2 z
= − iµ
= E
0
0
σE
0 r
8 δ 2 r 2
16 δ 4
(4 a 2
( r 2 − 2 a 2 ) ,
− r 2 E
2 z
(0) = −
E
0
16 δ 4
( a 2
(14)
− r 2 )(3 a 2 − r 2 ) , (15) since E
2 z
( a ) = 0 must hold. Now we calculate again Z = E
0
/I by adding the contribution of E
2 the current: to
Z −
1 = σ
Z a
0
1 − i r 2 −
2 δ 2 a 2
−
1
16 δ 4
( a 2 − r 2 )(3 a 2 − r 2 ) 2 πrdr = πa 2 σ 1 + ia 2
4 δ 2 a 4
−
12 δ 4
.
(16)
We thus find for the resistance
R = Re Z ≃ R
0 a 4
1 +
48 δ 4
.
(17)
We now apply the second method (“current” approach), in which the peak amplitude of the current is kept constant and equal to I
0
= πa 2 σE
0
. Thus, to the lowest order, using
I
0
.
σ
Z a
0
E
0
1 − ir 2
2 δ 2
E
1 z
(0) 2 πrdr , (18) we obtain
E
1 z
(0) = ia 2
4 δ 2
E
0
, E
0
E
1 z
( r ) ≃ E
0
1 − i
2 δ 2 r 2 − a 2
2
.
(19)
We thus calculate the dissipated power per unit length as
P d
=
Z a
0
1
2
σ | E
0
E
1 z
( r ) | 2 2 πrdr =
1
2
πa 2 σE 2
0 that yields
R = R
0 a 4
1 +
48 δ 4
, a 4
1 +
48 δ 4
1
2
σ R I 2
0
, (20)
(21) in agreement with the result obtained with the first method. Notice, however, that using the second method we did not need to compute the electric field up to the second order ( ∼ a 4 /δ 4 ) contribution.
We can also obtain the parasitic inductance using the above approach; in this case, according to the definition
L =
2
| I
˜
| 2
Z a
0
1
2 µ
0
B 2
1 φ
( r ) 2 πrdr , (22) that yields again L = µ
0
/ 8 π .
4

To calculate the impedance in the HF limit, first we notice that Eq.(9) has a rather simple exponential solution in planar geometry:
E z
= E
1 e
−
αx + E
2 e + αx , (23) with the coefficients determined by the boundary condition at the surface of the conductor and by the requirement that the field can not diverge deep inside the conductor. In the limit a ≫ δ , we may assume that the field also decays exponentially inside the cylindrical conductor (the effect of the curvature of the wire being negligible in this limit), so that we write the solution for the cylindrical wire for r < a as
E z
( r ) ≃ E
0 e α ( r
− a ) .
(24)
The impedance per unit length is thus given by
Z −
1 = σ
Z a
0 e α ( r
− a ) 2 πrdr ≃ 2 πσ aα − 1
α 2 a
≃ 2 πσ
α
, where we used δ ≪ a to let e
−
αa → 0. The real part of the impedance is given by
R ≃ (2 πσδa )
−
1 = R
0
( a/ 2 δ ) ,
(25)
(26) a result also quoted by Pauli 2 , without explicit derivation. This approximate relation shows that the current effectively flows through a surface ring with a thickness equal to δ and hence an area
≃ 2 πaδ . Also notice that in the HF limit, the real and imaginary part of the impedance are equal, thus the inductance L ≃ R /ω .
Now we consider the second method in the HF limit. The total current is given by
I
˜
=
Z a
0
σE
0 e α ( r
− a ) 2 πrdr = 2 πσE
0 a
α
−
1 − e
−
α 2
αa
≃ 2 πσE
0 a
α
, (27) so that | I
˜
| 2 = 2( πσa 2 | E
0
| ) 2 ( δ 2 /a 2 ). The dissipated power is given by
P d
=
1
2
Z a
σ | E
0 e α ( r
− a ) | 2 2 πrdr = πσ | E
0
| 2 e −
2 a/δ
Z a
0 0 e 2 r/δ rdr ≃ πσ | E
0
| 2 aδ
2
, (28) so that posing P d
= R| I
˜
| 2 / 2 one obtains again R = R
0
( a/ 2 δ ), in agreement with Eq.(26).
A simple lesson taken from this discussion is that the impedance of the system is well defined if the ends of system are connected either to an ideal AC voltage generator or to an ideal AC current generator, both defined as giving the same peak amplitude and cycle-averaged value of either voltage or current independently of the frequency. Nevertheless, the “ideal current generator” approach seems to be a bit more rigorous since the existence of a well defined I/O current at the ends of the system in a steady state is a consequence of the conservation of charge and is more “robust” with respect to boundary effects, i.e. to how contacts should be made to the external generator, or to the choice of end points for voltage measurements. Moreover, if the case of LF the “current” approach already provides the parasitic resistance at the lowest order in the inductive corrections to the electric field.
We also may notice that the name of “skin effect” is well justified in the HF regime, because the current is confined to the “skin” layer of thickness ∼ δ . In the LF regime the current is modulated in radial direction (so one may talk about “eddy currents” reducing the total current), but the effect can not simply be described as a reduction of the effective section of the wire.
2 W. Pauli, Electrodynamics , par.23 .
5