If the surface current source driving the conducting block of
Fig. 10.6.1 is a sinusoidal function of time

the current density tends to circulate through the block in the
neighborhood of the surface adjacent to the source. This tendency for
the sinusoidal steady state current to return to the source through
the thin zone or skin region nearest to the source gives another view
of magnetic diffusion.

To illustrate skin effect in specific terms we return to the
one-dimensional diffusion configuration of Sec. 10.6, Fig. 10.6.1.
Once again, the distributions of *H*_{z} and *J*_{y} are governed by the
one-dimensional diffusion equation and Ampère's law, (10.6.1) and
(10.6.2).

The diffusion equation is linear and has coefficients that are
independent of time. We can expect a sinusoidal steady
state response having the same frequency as the drive, (1). The
solution to the diffusion equation is therefore taken as having a
product form, but with the time dependence stipulated at the outset.

At a given location *x*, the coefficient of the exponential is a complex
number specifying the magnitude and phase of the field.

Substitution of (2) into the diffusion equation, (10.6.1),
shows that the complex amplitude has an *x* dependence governed by

where ^{2} j.

Solutions to (3) are simply *exp ( x)*. However,
is complex. If we note that *j = (1 + j)/2*, then it
follows that

In terms of the * skin depth* , defined by

One can also write (4) as

With *C*_{+} and *C*_{-} arbitrary coefficients, solutions to (3) are
therefore

Before considering a detailed example where these coefficients are
evaluated using the boundary conditions, consider the *x - t* dependence
of the field represented by the first solution in (7). Substitution
into (2) gives

making it clear that the field magnitude is an exponentially decaying
function of *x*. Within the envelope with the decay length
shown in Fig. 10.7.1, the field propagates in the *x* direction. That is,
points of constant phase on the field distribution have * t -
x/ = * constant and hence move in the *x* direction with the
velocity . Although the phase propagation signifies
that at a given instant, the field (and current density) are positive
in one region while negative in another, the propagation is
difficult to discern because the decay is very rapid.

**Figure 10.7.1** Magnetic diffusion
wave in the sinusoidal steady state, showing envelope with decay
length and instantaneous field at two different times. The
point of zero phase propagates with the velocity .
The second solution in (7) represents a similar diffusion wave,
but decaying and propagating in the *-x* rather than the *+x* direction.
The following illustrates how the two diffusion waves combine to
satisfy boundary conditions.

#### Example 10.7.1. Diffusion into a Conductor of Finite Thickness

We consider once again the field distribution in a conducting
material sandwiched between perfectly conducting plates, as shown in
either Fig. 10.7.2 or Fig. 10.6.1. The surface current density of the
drive is given by (1) and it is assumed that any transient
reflecting the initial conditions has died out. How does the
frequency dependence of the field distribution in the conducting block
reflect the magnetic diffusion process?

**Figure 10.7.2** (a) One-dimensional magnetic
diffusion in the sinusoidal steady state in the same configuration as
considered in Sec. 10.6. (b) Distribution of the magnitude of *H*_{z}
in the conducting block of (a) as a function of the skin depth.
Decreasing the skin depth is equivalent to raising the frequency.
Boundary conditions on *H*_{z} are the same as in Sec. 10.6,
*H*_{z} (-b, t) = K_{s}(t) and *H*_{z} (0, t) = 0. These are satisfied by
adjusting the complex amplitude so that

It follows from (7) that the second of these is satisfied if
*C*_{+} = -C_{-}. The first condition then serves to evaluate *C*_{+} and
hence *C*_{-}, so that

This expression represents the superposition of fields propagating
and decaying in the * x* directions, respectively. Evaluated at a
given location *x*, it is a complex number. In accordance with (2),
*H*_{z} is the real part of this number multiplied by *exp (j
t)*. The magnitude of *H*_{z} is the magnitude of (10), and is shown
with the skin depth as a parameter by Fig. 10.7.2.

Consider the field distribution in two limits. First, suppose
that the skin depth is very large compared to the thickness *b* of the
conducting block. This might be the limit in which the frequency is
made very low compared to the reciprocal magnetic diffusion time based
on the conductor thickness.

In this limit, the arguments of the exponentials in (10) are small.
Using the approximation *exp (u) 1 + u*, (10) becomes

Substitution of this complex amplitude into (2) gives the space-time
dependence.

The field has the linear distribution expected if the current density
is uniformly distributed over the length of the conductor. In this
large skin depth limit, the field and current density spatial
distributions are essentially the same as if the current source were
time independent.

In the opposite extreme, the skin depth is short compared to the
conductor length. Perhaps this is accomplished by making the
frequency very high compared to the reciprocal magnetic diffusion time
based on the conductor length.

Then, the first term in the denominator of (10) is large compared with
the second. Division of the numerator by this first term gives

In justifying the second of these expressions, remember that *x* is
negative throughout the region of interest. Substitution of (15)
into (2) shows that in this short skin depth limit

With the origin shifted from *x = 0* to *x = -b*, this field has the
*x - t* dependence of the diffusion wave represented by (8). So it
is that in the short skin depth limit, the distribution of the field
magnitude shown in Fig. 10.7.2 has the exponential decay typical of
skin effect.

The skin depth, (5), is inversely proportional to the square
root of . Thus, an order of magnitude variation in
frequency
or in conductivity only changes by about a factor of about 3.
Even so, skin depths found under practical conditions are widely
varying because these parameters have enormous ranges. In good
conductors, such as copper or aluminum, Fig. 10.7.3 illustrates how
varies from about 1 cm at 60 Hz to less than 0.1 mm at l MHz.
Of interest in determining magnetically induced currents in flesh is
the curve for skin depth in materials having the "physiological"
conductivity of about 0.2 S/m (Demonstration 7.9.1).

**Figure 10.7.3** Skin depth as a
function of frequency.
If the frequency is high enough so that the skin depth is small
compared with the dimensions of interest, then the fields external to
the conductor are essentially determined using the perfect
conductivity model introduced in Sec. 8.4. In Demonstration 8.6.1,
the fields around a conductor above a ground plane line were derived
and the associated surface current densities deduced. If these
currents are in the sinusoidal steady state, we can now picture them
as actually extending into the conductors a distance that is on the
order of .

Although skin effect determines the paths of
current flow at radio frequencies, as the following demonstrates, it
can be important even at 60 Hz.

The core of magnetizable material shown in Fig. 10.7.4 passes
through a slit cut from an aluminum block and through a winding that
is driven at a frequency in the range of 60-240 Hz. The winding and
the block of aluminum, respectively, comprise the primary and secondary
of a transformer. In effect, the secondary is composed of one turn
that is shorted on itself.

**Figure 10.7.4** Demonstration of skin effect.
Currents induced in the conducting block tend to follow paths of
minimum reactance nearest to the slot. Thus, because the aluminum
block is thick compared to the skin depth, the field intensity
observed decreases exponentially with distance *X*. In the
experiment, the block is 10 x 10 x 26 cm with thickness
of 6 cm between the right face of the slot and the right side of the
block. In aluminum at 60 Hz, = 1.1 cm, while at 240 Hz
is half of that. To avoid distortion of the field, the yoke
is placed at one end of the slot.
The thickness *b* of the aluminum block is somewhat larger than a
skin depth at 60 Hz. Therefore, currents circulating through the block
around the leg of the magnetic circuit tend to follow the paths of
least reactance closest to the slit. By making the length of the
block and slit in the *y* direction large compared to *b*, we expect to
see distributions of current density and associated magnetic field
intensity at locations in the block well removed from the ends that
have the *x* dependence found in Example 10.7.1.

In the limit where is small compared to *b*, the magnitude of
the expected magnetic flux density *B*_{z} (normalized to its value
where *X = 0*) has the exponential decay with distance *x* of the inset to
Fig. 10.7.4. The curves shown are for aluminum at frequencies of
60 Hz and 240 Hz. According to (5), increasing the frequency by a
factor of 4 should decrease the skin depth by a factor of 2. Provision
is made for measuring this field by having a small slit milled in the
block with a large enough width to permit the insertion of a
magnetometer probe oriented to measure the magnetic flux density in
the *z* direction.

As we have seen in this and previous sections, currents induced
in a conductor tend to exclude the magnetic field from some region.
Conductors are commonly used as shields that isolate a region from its
surroundings. Typically, the conductor is made thick compared to the
skin depth based on the fields to be shielded out. However, our
studies of currents induced in thin conducting shells in Secs. 10.3
and 10.4 make it clear that this can be too strict a requirement for
good shielding. The thin-sheet model can now be seen to be valid if
the skin depth is * large* compared to the thickness of the
sheet. Yet, we found that for a cylindrical shell of radius *R*,
provided that * R 1*, a sinusoidally varying
applied field would
be shielded from the interior of the shell. Apparently, under certain
circumstances, even a conductor that is thin compared to a skin depth
can be a good shield.

**Figure 10.7.5** Perfectly
conducting -shaped conductors are driven by a distributed
current source at the left. The magnetic field is shielded out of
the region to the right enclosed by the perfect conductors by: (a) a
block of conductor that fills the region and has a thickness *b* that
is large compared to a skin depth; and (b) a sheet conductor having a
thickness that is less than the skin depth.
To understand this seeming contradiction, consider the
one-dimensional configurations shown in Fig. 10.7.5. In the first of
the two, plane parallel perfectly conducting electrodes again
sandwich a block of conductor in a system that is very long in a
direction perpendicular to the paper. However, now the plates are
shorted by a
perfect conductor at the right. Thus, at very low frequencies, all of
the current from the source circulates through the perfectly
conducting plates, bypassing the block. As a result, the field
throughout the conductor is uniform. As the frequency is raised, the
electric field generated by the time-varying magnetic flux drives a
current through the block much as in Example 10.7.1, with the current
in the block tending to circulate through paths of least reactance
near the left edge of the block. For simplicity, suppose that the
skin depth is shorter than the length of the block *b*, so that
the decay of current density and field into the block is essentially
the exponential sketched in Fig. 10.7.5a. With the frequency high
enough to make the skin depth short compared to *b*, the field tends to
be shielded from points within the block.

In the configuration of Fig. 10.7.5b, the block is replaced by a
sheet having the same and but a thickness that is
less than a skin depth . Is it possible that this thin sheet
could suppress the field in the region to the right as well as the
thick conductor?

The answer to this question depends on the location of the
observer and the extent *b* of the region with which he or she is
associated. In the conducting block, shielding is poor in the
neighborhood of the
left edge but rapidly improves at distances into the interior that are
of the order of or more. By contrast, the sheet conductor can
be represented as a current divider. The surface current, *K*_{s}, of
the source is tapped off by the sheet of conductivity per unit width
*G = /h* (where *h* is the height of the structure)
connected to the inductance (assigned to unit width) *L = bh* of
the single-turn inductor. The current through the single-turn
inductor is

This current, and the associated field, is shielded out effectively
when *| LG| = b 1*.
With the sheet, the shielding strategy is to make equal use of all of
the volume to the right for generating an electric field in the sheet
conductor. The efficiency of the shielding is improved by making
* b* large: The interior field is made small by
making the shielded volume large.