Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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394 Time Dependent Problems I (Quasi Stationary Approximation)

(6.250)

The inverse transform by means of the residue theorem occurs in similar manner as

before for the functions . Recognizing the additional pole at

and the additional factor in the denominator gives

(6.251)

and

(6.252)

Another avenue is to start with the general solution (6.245) to obtain

(6.253)

We have used the fact that

(6.254)

This is the Fourier-Bessel series for the function 1 (the number one) in the range

. We shall prove this using the following Ansatz:

We multiply it with and integrate over

x from 0 to 1:

xp,()

xi p()

ip()

-----------------------------=

xi p()J

ip()⁄

–=

x τ,()B

x() λ

– τ()exp

()

-------------------------------------------------------

n 1=

∞

∑

–=

x τ,()

-----------

x∂

∂B

–=

-----------

x() λ

– τ()exp

()

----------------------------------------------------

n 1=

∞

∑

–=

x τ,()B

2λ

x() λ

– τ()exp

()

----------------------------------------------------------------

()exp 1–

--------------------------------------

n 1=

∞

∑

x()

()

----------------------------

n 1=

∞

∑

x() λ

– τ()exp

()

-------------------------------------------------------

n 1=

∞

∑

–=

x τ,()B

x() λ

– τ()exp

()

-------------------------------------------------------

n 1=

∞

∑

– .=

x()

()

----------------------------

n 1=

∞

∑

0 x 1<≤

1 c

x()

n 1=

∞

∑

0n'

x()

6.7 The Cylindrical Diffusion Problem 395

The last step is a result of the orthogonality relation (3.214). Furthermore, using

our little set of formulas from Sect. 3.7.2 page 170 we find

The coefficients are therefore

This proves our claim. The results of (6.251) and () agree. The expansion (6.254)

makes the result (6.251) understandable. According to (6.251) for one has

just as the initial condition requires.

The exponential terms may be neglected for very large times, which results in

Of course, this is necessary. For very large times, the uniform magnetic field

occupies the entire space, including the inside of the cylinder. The currents which

initially shielded the inside of the cylinder from the magnetic field decay over time.

The time to decay is of the order of magnitude

(6.255)

Disregarding the factor this agrees with our rough approximation (6.39).

6.7.3 The azimuthal Field B

We will now discus the azimuthal field in a similar manner as the longitudinal

field. Now, we create a magnetic field

(6.256)

at the surface of a cylinder, while there is no initial field inside of the cylinder.

Then eq. (6.224) applies to . This results for in

0n'

()xd

∫

n 1=

∞

∑

x()J

0n'

()xd

∫

-----

n 1=

∞

∑

()[]

nn'

------

0n'

()[]

.==

0n'

x()xd

∫

0n'

()

0n'

--------------------=

0n'

()

-----------------------------=

τ 0=

x 0,()B

x()

()

----------------------------

n 1=

∞

∑

– B

11–[]0===

x τ,()[]

τ∞→

κr

---------------

1≈=

µκr

------------

≈

µκr

2.40()

------------------=

2.40

()

x τ,()[]

x 1=

1 τ,()f

τ()==

x τ,() B

xp,()

396 Time Dependent Problems I (Quasi Stationary Approximation)

(6.257)

The boundary conditions are

(6.258)

. (6.259)

The only difference to the longitudinal case is that now, replaces . The

solution is obtained in analogy to (6.239)

(6.260)

For the inverse transform, we need the residues of the function

. (6.261)

Its poles are at

(6.262)

(6.263)

The zero of is no pole but a removable singularity, since the

numerator vanishes there as well. The ratio remains finite

(= x). The residues are then

Using the identity

and

gives

, (6.264)

and similar to (6.244), the transformation is

---

x∂

∂

x∂

∂

-----– p–





xp,()0=

1 p,()f

p()=

0 p,()finite=

xp,()f

p()

xi p()

ip()

----------------------

xi p()

ip()

----------------------

pτ()exp

ip λ

= n 1≥()

–= n 1≥()

0= J

ip()

xi p()J

ip()⁄

x() λ

– τ[]exp

ip()

----------------------------------------------------

p λ

–→

lim p λ

+()=

x() λ

– τ[]exp

2 p

----------

J '

ip()

----------------------------------------------------

p λ

–→

lim .=

J '

z() J

z()

---

z()–=

J '

()J

()=

2λ

x() λ

– τ[]exp

()

----------------------------------------------------------------–=

6.7 The Cylindrical Diffusion Problem 397

(6.265)

Applying the convolution theorem to eq. (6.260) yields

(6.266)

As an example, let us consider a similar case as before, now with the

boundary condition

(6.267)

(6.268)

Then we write the field with (6.260)

(6.269)

The function

has poles at and also at . The residue there can be found most

conveniently by finding the initial term of the expansion of . According to

(3.174) for small magnitudes of the argument it is approximately

and thus, for the residue there

(6.270)

The remaining residues are found similarly to (6.264), now with an additional

factor in the denominator.

(6.271)

Thus

2λ

x() λ

– τ[]exp

()

----------------------------------------------------------------

n 1=

∞

∑

–

xi p()

ip()

----------------------

x τ,()

2λ

x() λ

– τ[]exp

()

----------------------------------------------------------------

() λ

()exp τ

∫

n 1=

∞

∑

–=

τ() B

p() B

p⁄=

xp,()

xi p()

ip()

-----------------------------=

xi p()

ip()

-----------------------------

pτ()exp

–=

xi p()

xi p

------------

≈

xi p()

ip()

-----------------------------

pτ()pexp

p 0→

lim B

xi p

------------

---------

------------

x===

–=

x() λ

– τ[]exp

()

-------------------------------------------------------

398 Time Dependent Problems I (Quasi Stationary Approximation)

(6.272)

The same result can be derived using the general solution (6.266)

(6.273)

The Fourier-Bessel series for x in the range has to be used here:

(6.274)

The proof is achieved using the formulas from Sect. 3.7.3.3 and is left for the

reader as an exercise. Thus, both methods provide the same result. With (6.272)

and (6.274) follows for the time

that is, the initial condition for a vanishing field inside the cylinder is satisfied,

indeed. For very large times the field becomes as required:

(6.275)

Prescribing on the boundary of the cylinder means that the total current I inside

the cylinder is determined:

(6.276)

The current I remains constant during the entire diffusion process, merely the

current density varies over time. Initially, the current flows entirely in the

cylinder surface, and finally, the current has a uniform density inside the entire

cylinder.

The field that linearly increases with the radius (6.275) corresponds to this final

state. One can also derive the steady state directly from (6.217), from which

follows for the steady state that

(6.277)

x τ,()B

x() λ

– τ[]exp

()

-------------------------------------------------------

n 1=

∞

∑

x τ,()

x() λ

– τ()exp

()

-----------------------------------------------------------------------

τ[]exp 1–

-----------------------------------







n 1=

∞

∑

–=

x()

()

--------------------------------

n 1=

∞

∑

–

x() λ

– τ[]exp

()

--------------------------------------------------------------

n 1=

∞

∑

x() λ

– τ[]exp

()

-------------------------------------------------------

n 1=

∞

∑

0 x 1<≤

x()

()

----------------------------

n 1=

∞

∑

–=

τ 0=

x 0,()B

x()

()

----------------------------

n 1=

∞

∑

+ B

xx–[]0===

x 0,()[]

τ∞→

----

r()2πrrd

∫

------

2πr

rt,()

---

r∂

∂

r∂

∂

-----–





r() 0=

6.7 The Cylindrical Diffusion Problem 399

with the general solution

(6.278)

To avoid the divergence at requires that

and the boundary condition at is satisfied if

Therefore, , as claimed.

6.7.4 Skin Effect in a Cylindrical Wire

As another special case of diffusion, we discuss the skin effect of a cylindrical wire

in an azimuthal field . There shall be no initial field inside the wire. The

current

(6.279)

shall start flowing through the wire at time . At the surface, this

creates the field

(6.280)

. (6.281)

Thus, the boundary conditions in the time and p-domain, respectively are

(6.282)

. (6.283)

With (6.260), we obtain

(6.284)

The function

(6.285)

has poles of order 1 at

(6.286)

(6.287)

r() Ar

---+=

r 0=

B 0=

⁄=

x τ,()

Ωτ()cos=

t 0 τ 0=()=

2πr

-----------

Ωτ()cos B

Ωτ()cos==

2πr

-----------=

τ() B

Ωτ()cos=

p() B

Ω

-------------------

xp,()

Ω

-------------------

xi p()

ip()

----------------------

xi p()

piΩ+()piΩ–()J

ip()

-------------------------------------------------------------==

xi p()

piΩ+()piΩ–()J

ip()

-------------------------------------------------------------

pτ()exp

ip λ

–=

400 Time Dependent Problems I (Quasi Stationary Approximation)

and also at

(6.288)

Its residues at the locations are

(6.289)

This can be obtained from the residues (6.264) and the factor

The residues of the poles at are

(6.290)

Therefore

(6.291)

Again, the exponentially decaying terms are insignificant for large times. This

gives the “steady state” for which one obtains

(6.292)

Particularly for , in harmony with the boundary condition, the field

becomes

The functions are complex valued, however, one may split them into

their real and imaginary parts. These functions have such great importance that

new functions and names were introduced to address them, the Kelvin functions

with “ber” (Bessel real part) and “bei” (Bessel imaginary part)

(6.293)

iΩ±=

–=

()

Ω

+()J

()

-----------------------------------------------

τ–()exp=

Ω

-------------------

Ω

----------------------–=

iΩ±=

iΩ±()J

xi iΩ±()

2 iΩ±()J

iiΩ±()

---------------------------------------------------

iΩ±τ()exp=

------

xi iΩ±()

iiΩ±()

-----------------------------

iΩ±τ()exp=

x τ,()

------

xiΩ–()

iΩ–()

---------------------------

iΩτ()exp

xiΩ()

iΩ()

------------------------

iΩ– τ()exp+=

2λ

x()

Ω

+()J

()

-----------------------------------------------

τ–()exp

n 1=

∞

∑

x τ,()

------

xiΩ–()

iΩ–()

---------------------------

iΩτ()exp

xiΩ()

iΩ()

------------------------

iΩ– τ()exp+=

x 1 rr

=(),=

1 τ,()

------

iΩτ()exp iΩ– τ()exp+[]B

Ωτ()cos==

xi iΩ

−

()

xiΩ

−

()J

x Ω i

−

()ber

x Ω()i bei

x Ω()±==

6.7 The Cylindrical Diffusion Problem 401

Kelvin functions allow one to write as of (6.291) or (6.292) in a real

form. We will leave the details for the reader’s further study. Here, we will suffice

with a short discussion of two limits (very low and very high frequencies).

1) The limit of very low Frequencies ( )

If , then the magnitude of all arguments of the Bessel functions in (6.292) are

much smaller than 1 because x is in the interval . Consequently

(6.294)

and therefore

(6.295)

This should come as no surprise. The field is in phase everywhere because we

assumed that the frequency of the field is small and its corresponding period is

consequently large, that is, it is large against the time it takes to penetrate the

cylinder. From

(6.296)

follows that

(6.297)

Note eq. (6.137) with . The amplitude increases linearly with x (i.e., linear

with the radius). Therefore, the current density inside the cylinder is spatially

constant for all times. However, it oscillates over time with the period . In

principle, this is the behavior of a direct current, which changes only slowly

(slowly compared to the typical penetration depths, ).

2) The limit of very high frequencies ( )

A zero order approximation for very large arguments is this:

(6.298)

. (6.299)

Using this in (6.292) and (6.293) gives

x τ,()

Ω 1«

0 x 1<≤

xiΩ±()

iΩ±()

---------------------------

x≈

x τ,()

------

xiΩτ()exp xiΩ– τ()exp+[]≈ B

x Ωτ()cos=

Ωωµκr

1«=

---- µκr

Ωτ()cos

µκr

Ω 1»

ber

x Ω()

Ω

----





Ω

----

3π

------+





cosexp

2πx Ω

-------------------------------------------------------------------

≈

bei

x Ω()

Ω

----





exp x

Ω

----

3π

------+





sin

2πx Ω

------------------------------------------------------------------

≈

402 Time Dependent Problems I (Quasi Stationary Approximation)

(6.300)

The reader is cautioned that this expression is valid only for (that is, even

for large frequencies, its validity is restricted to x being not too small, i.e., not too

close to the axis). Take eq. (6.300), close to the surface where or ,

there one observes that the field behaves just like the planar case. For this purpose,

compare the current result (6.300) with that of the half-space (6.139) and notice

that there, the distance ξ from the surface of the half space corresponds here to the

distance from the cylinder surface , or dimensionless:

. From

(6.301)

follows that

(6.302)

that is, the penetration depth is very small versus the cylinder radius. Also plausible

is, that under those circumstances the diffusion occurs as in the plane case. This is

true, not only for cylinders, but for all kinds of shapes, as long as the frequency is

large enough, or the interest is only in sufficiently thin penetration depths. From

this perspective, the result (6.139) is of rather general significance.

Summarizing, we may conclude that the case of very low frequencies can be

reduced to the case of direct currents, while the case of high frequencies can be

reduced to a plane diffusion problem. For intermediate frequencies, there is no easy

approximation. However, the behavior is qualitatively (not quantitatively) similar

to the plane case that we have studied in Sect. 6.5.4: The wave is damped while

penetrating the medium and it exhibits a phase difference.

6.8 Limits of the Quasi Stationary Theory

The quasi stationary theory is an approximation which is based on neglecting the

displacement current in Maxwell’s equations. We have mentioned already that all

phenomena related to electromagnetic waves are neglected. It may sound paradox

that we have encountered wave behavior in processes like the skin effect. Notice

however, that these processes are enforced by the boundary conditions and are

unrelated to electromagnetic waves which we will discuss later.

A typical behavior of propagation of waves is that this occurs with a certain

finite velocity. We will discuss this in detail in the next chapter. Furthermore, the

fundamental postulate of relativity, and thereby for the entire natural science

altogether, is that there is no signal velocity higher than the speed of light in

vacuum, . Consider for example, a field of the kind as discussed in

Sect. 6.4, initially shaped like a δ-function, propagating in the infinite space.

x τ,()

1 x–()–

Ω

----exp Ωτ 1 x–()–

Ω

----





cos

------------------------------------------------------------------------------------------------------

≈

x Ω 1»

≈ x 1≈

r–

r–()r

⁄ 1 rr

⁄–()1 x–==

ωµκr

Ω 1»=

ωµκ

---------------»

310

1–

⋅≈

6.8 Limits of the Quasi Stationary Theory 403

Eq. (6.69) describes such a field. This equation indicates something very peculiar.

At the time , the field exists only at one particular place, however, it is there

infinitely large. Then the field already penetrates the entire space after an

arbitrarily small period of time , even though it is very small at large distances.

This seems to suggest that the signal velocity is infinitely large. Indeed, given

sufficiently sensitive instruments, it should be possible to detect a field at very

large distances after an extremely short time period, and use this to transmit

signals, at least in principle, if these fields actually existed. Nonetheless, such is not

the case. The infinite signal speed is typical for “diffusion processes”, i.e., for those

processes described by the diffusion equation. However, physically they are not

real. Formally – as we find – this results from neglecting the displacement current,

which is equivalent to the assumption of infinite signal propagation velocity.

Another example we have dealt with shall be mentioned: The conductive

half-space with no initial field. When a constant field is suddenly applied to the

surface at the time , then according to eq. (6.117) or Fig. 6.20, this field

penetrates the entire half-space after an arbitrarily short time period.

All this should not be interpreted in a manner, rendering all these fields as

useless or totally wrong. Under appropriate preconditions, the quasi stationary

theory provides an excellent approximation of the actual field behavior. The

enormous velocity of the speed of light is thereby fundamental. The fields

propagate with this speed, and the far away areas which have not been reached by

the field are for many problems immaterial. Besides, even though these faraway

fields described by the quasi stationary theory do not vanish, nevertheless, they are

oftentimes so small that they are deemed to be insignificant. The qualitative

conclusion of these remarks is that the quasi stationary theory is a useful

approximation only for sufficiently large times or sufficiently slow or low

frequency processes. To arrive at a quantitative statement, we return to the example

of the field diffusion in the infinite space. In contrast to eq. (6.69), the field has to

vanish if

In practice, this is insignificant if

that is, if

Therefore, it must be:

τ 0=

x' x–()

ξ' ξ–()

µε

------==

ξ' ξ–()

4τ

--------------------–exp

x' x–()

-------------------

– µκexp 1«= for x' x–()

µε

------=

x' x–()

µκ

-------

1» for x' x–()

µε

------=

µκ

µε

-------

t 1»