An Analog of Time-domain Kirchhoff Formulas for the Electromagnetic Field in a Conducting Medium

Studies in System Science (SSS) Volume 1 Issue 1, March 2013

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An Analog of Time-domain Kirchhoff Formulas for the Electromagnetic Field in a Conducting Medium Anatolii S. Il’inskii, Irina G. Efimova Faculty of Computational Mathematics and Cybernetics, Moscow State University Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991 Russia irina.efimova@mail.ru Abstract An analog of Kirchhoff formulas is obtained in the time domain for the vectors of the intensities of the nonstationary electromagnetic field in a conducting medium. At any instant, the field at an arbitrary point in a closed volume is represented as the sum of an integral over the surface bounding this volume and an integral over the volume. The integrands contain field intensities, external currents, and their time derivatives at instants preceding the observation instant.

Keywords Time Domain; Kirchhoff Conducting Medium

Formulas;

Electromagnetic Field;

Introduction It is well known that a monochromatic electromagnetic field in a homogeneous isotropic medium can be expressed in an integral form through the values of the vectors of the electric- and magnetic-field intensities on a closed surface [1-3]. The frequency-domain integral representations of the electric and magnetic fields referred to as the Stratton--Chu formulas are valid for any values of the constitutive parameters of the medium. These formulas are obtained and thoroughly analyzed in [1]. The Stratton--Chu representations are widely applied for deriving integral equations that are used for solution of various electrodynamic problems (see, e.g., [2-6]).

In the time domain, the electromagnetic field existing in a nonconducting medium also can be represented in an integral form through the intensities of the electric and magnetic fields on a closed surface. In this case, the field intensities entering the integrands are functions of a retarded time argument. Such representations are obtained in scalar [1] and vector [2] forms and are successfully applied to derive integral equations used in simulations of scattering of a

nonstationary electromagnetic field by perfectly conducting or lossless dielelectric bodies located in nonconducting media [2, 7-9]. However, the method of time-domain integral equations has not been developed for the case of a conducting medium, because an integral representation for a nonsstationary electromagnetic field in such a medium has not been obtained. In monograph [1], it is noted that, when the Kirchhoff method is applied for integration of the inhomogeneous wave equation, “The presence of conductivity introduces serious analytical difficulties”.

The purpose of this study is to obtain a time-domain integral representation of a nonstationary electromagnetic field in a homogeneous isotropic medium with conductivity.

A Form of the Wave Equation Suitable for Obtaining Integral Formulas for Its Solution in a Conducting Medium Consider a homogeneous isotropic medium with constitutive parameters that do not depend on either time or spatial coordinates. In this case, the Maxwell equations have the form

    ∂H rotE + µ = − j mext − σ m H ∂t ,    ∂E  eext e rotH − ε = j +σ E ∂t

 E

 H

are the vectors of the electric- and e magnetic-field intensities, respectively; ε , µ , σ , where

and

(1)

m and σ are the medium permittivity, permeability, specific electric conductance, and specific magnetic  eext  mext conductance, respectively; and j and j are the volume densities of external electric and magnetic

1

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Studies in System Science (SSS) Volume 1 Issue 1, March 2013

currents, respectively. It is assumed that

 E

ε , µ , σ e,  H are finite

and and σ are constants and that functions continuous along with all of their derivatives at all ordinary spatial points (where the medium does not abruptly change its physical properties).

m

We assume that the external currents and charges satisfy the continuity equations

  ∂ρ mext ∂ρ eext = 0, = 0 , divj mext + divj eext + ∂t ∂t where ρ

eext

and

ρ mext

(2)

are the volume densities of

the external electric and magnetic charges, respectively.

We assume in addition that all functions of time that are considered in this study---the electric-and magnetic-field intensities and the volume densities of external currents and charges---are zero along with their derivatives till the instant t = 0 inclusively.

Maxwell equations (1) and continuity equations (2) imply the relationships

t ∂ρ eext (t )  1 be be exp( e t )dt , divE = e exp(− e t ) ∫ ∂t a a a t =0

 divH

=

t ∂ρ mext (t ) 1 bm bm exp( ) exp( − t t )dt , ∫ ∂t am am t = 0 am

(3)

where

a e = ε , be = σ e , a m = µ, bm = σ m .

(4)

Equations (1) and (3) are the complete system of equations describing the electromagnetic field in a homogeneous isotropic medium. Let us present the





vector wave equation for E and H . We have it from the first and second Maxwell equations

  ∂ E rotrotE + εµ 2 ∂t     ∂j eext e ∂E = −µ − µσ − rotj mext − σ m rotH ∂t ∂t 2

and   ∂2H rotrotH + εµ 2 ∂t

2

(5)

    ∂j mext ∂ H = −ε − εσ m + rotj eext + σ e rotE , ∂t ∂t

(6)

respectively. Let us modify wave equations (5) and (6) into the form

     ∂2E ∂E e ∂E m e ) rotrotE + εµ 2 + µσ + σ (σ E + ε ∂t ∂t ∂t

   ∂j eext = −µ − rotj mext − σ m j eext ∂t

(7)

     ∂2H ∂H m ∂H e m − σ (−σ H − µ rotrotH + εµ 2 + εσ ) ∂t ∂t ∂t

   ∂j mext = −ε + rotj eext − σ e j mext ∂t

(8)



Below, we perform calculations only for field E and



obtain the result for field H according to the duality principle. By virtue of the vector identity

   rotrotE = graddivE − ∆E ,

equation (7) can be represented as

   1 ∂2E  E ∂ ∆E − 2 2 − ( µσ e + εσ m ) − σ mσ e E ∂t c ∂t  eext  ∂j = graddivE + µ + σ m j eext + rotj mext , ∂t

where c =

1

εµ

(9)

is is the velocity of light. Let us

introduce the notation

a = µσ e + εσ m , b = σ mσ e .

(10)

Then, the wave equation in forms (7) and (9) can be represented as

   1 ∂2E  ∂E rotrotE + 2 2 + a + bE ∂t c ∂t    ∂j eext = −µ − rotj mext − σ m j eext , ∂t

   1 ∂2E  ∂E ∆E − 2 2 − a − bE ∂t c ∂t

(11)

Studies in System Science (SSS) Volume 1 Issue 1, March 2013

  ∂j eext = graddivE + µ + σ m j eext + rotj mext , ∂t

(12)

respectively. The wave equation can be simplified through eliminating the first-order time derivative of the field with the help of the replacement

 ~ E = E exp(−ac 2 t / 2) .

(13)

In addition, we make the following replacements in equation (7):

 eext ~ j = exp(−ac 2 t / 2) j eext ,  mext ~ j = exp(−ac 2 t / 2) j mext .

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arbitrary direction and function

ϕ

will be chosen

below. Using formulas of vector analysis, we obtain (16) in the form

     ∫ ϕrotrotE + E∆ϕ + gradϕdivE dv − ∫ (n ⋅ E ) gradϕds = V S

[

]

    = ∫ ([ gradϕ × [ E × n ]] − ϕ [rotE × n ])ds .

S

(18)

Introducing the replacement

 ~ H = H exp(−ac 2t / 2)

(19)

~

(14)

Calculating the first-and second-order time derivatives of the field and taking into account formulas (13) and (14), we modify equation (7) into the form

~ ~ ∂ 2 E ~ rotrotE + 2 2 + (b − a 2c 2 / 4) E c ∂t ~  ~ ~ ∂j eext ac 2 ~ = − µ ( ∂t − 2 j eext ) − rotj mext − σ m j eext .

(15)

and substituting the expression for rotrotE from (15) into (18) with allowance for (13) and (14), we obtain ~ ~ 1 ∂2E ∂j eext ac 2 ~ eext a 2 c 2 ~ ~ mext )E − [ j − j ) − ϕ 2 2 − ϕ (b − × gradϕ ] ∫ [−ϕµ ( 2 4 ∂t c ∂t V ~ ~ ~ − ϕσ m j eext ]dv + ∫ E∆ϕdv + ∫ gradϕdivEdv) V V

~   ~ = ∫ (n ⋅ E ) gradϕds + ∫ [ gradϕ × [ E × n ]]ds S S ~ ~ ∂H ac 2 ~  . − ∫ ϕ [( − σ m H − µ H ) × n ]ds +µ ∂t 2 S

Final Integral Formulas

Let

In order to derive an integral representation of the intensity of the nonstationary electric field satisfying wave equation (15), we apply the technique from monograph [1] and use the vector analog of the Green’s theorem. Let V be a closed spatial region bounded by regular (satisfying the Lyapunov

integrals from formula (20), respectively; i.e.,

conditions) surface

  S , Q and P

be vector functions

of a point that, along with their first- and second-order derivatives, are continuous in region

V

and on

surface S . Then, as shown in (1), the vector analog of the Green’s theorem

    ∫ (QrotrotP −P rotrotQ)dv V

     (16) = ∫ [ P rotQ] − [Q rotP] n ds , S  where n is the outward normal to surface S , is valid.

(

)

Let us set in (16)

    Q = ϕα and P = E , 

where

α

(17)

is a constant unit vector oriented in an

IV

and

IS

(20)

denote the volume and surface

~ ~ ∂j eext ac 2 ~ eext 1 ∂2E IV = ∫ [−ϕµ ( − j ) −ϕ 2 2 ∂ t 2 c ∂t V

a 2c 2 ~ ~ mext )E − [ j − ϕ (b − × gradϕ ] 4 ~ ~ ~ − ϕσ m j eext ]dv + ∫ E∆ϕdv + ∫ gradϕdivEdv) , V V ~   ~ I S = ∫ (n ⋅ E ) gradϕds + ∫ [ gradϕ × [ E × n ]]ds − S S

~ ~ ~  ∂H − ∫ ϕ [(−σ H − µ + µac 2 / 2 H ) × n ]ds . ∂t S m

As

ϕ , we choose a solution to the scalar homogeneous

wave equation for a nonconducting medium

∆ϕ −

1 ∂ 2ϕ c 2 ∂t 2

=0

3

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Studies in System Science (SSS) Volume 1 Issue 1, March 2013

and calculate the sum We have

~ ~ 1 ∂2E Σ1 = E∆ϕ − ϕ 2 2 . c ∂t

to the limit as (21)

IV

~ ~ − ϕσ m j eext + gradϕdivE ]dv) .

γ →0

. Later, we will integrate the

left- and right-hand sides of equality (20) with respect to time from t = −∞ to t = ∞ and pass to the limit as

γ →0

.

(22)

1 1 1 (t + t ′) 2 1 exp(− ) = D0 (t + t ′) , (23) 2 r 2π γ r 2γ

r = ( x′ − x) 2 + ( y ′ − y ) 2 + ( z ′ − z ) 2 , ( x′, y ′, z ′) are the Cartesian coordinates of the observation point, ( x, y , z ) are the Cartesian

where

coordinates of the integration point in formula (16), is a certain small parameter that,

∞

1 ~ ′ 1 ∂ ~ ∂ ( r D0 (t + t )) 1 ∂E + 2 (E − D0 (t + t ′) ) r ∂t ∂t c ∂t 1 a 2c 2 ~ )E − D0 (t + t ′)(b − 4 r ~ ~ 1 1 1 ∂D  − [ j mext × ( D0 − 0 )r0 ] − D0 (t + t ′)σ m j eext + r r r ∂r

~ ∂D  1 1 (26) + ( D0 − 0 )r0 divE ]dvdt , r r ∂r  where r0 is the unit vector oriented from integration point ( x, y , z ) to observation point ( x ′, y ′, z ′) .

By virtue of (24), the order of integration in (26) can be changed. Since, according to (23),

later, will be let go to zero.

t = −∞

We consider continuous functions such that

~ 1 ∂ ~ ∂ϕ ∂E − ϕ )dt ∫ 2 (E ∂t ∂t t = −∞ c ∂t

(24)

V t = −∞

and ∞

∞

lim ∫ ∫ dvdt = ∫ lim ∫ dtdv .

γ → 0 V t = −∞

4

V γ → 0 t = −∞

and

t = ∞ , we have

ϕ=

∂ϕ =0 ∂t

when

∞

∞

t = −∞ V

into volume

~ 1 ∂j eext ac 2 ~ eext ∫ IV dt = ∫ ∫ [− r D0 (t + t′)µ ( ∂t − 2 j ) t = −∞ t = −∞V ∞

Following (1), we set

∞ dvdt = ∫ ∫ ∫ ∫ dtdv

ϕ

integral (22) and integrate the result with respect to time from t = −∞ to t = ∞ . As a result, we obtain

~ a 2c 2 ~ 1 ∂ ~ ∂ϕ ∂E )E + 2 (E − ϕ ) − ϕ (b − 4 ∂t ∂t c ∂t ~ − [ j mext × gradϕ ]

γ

-

is normalized so that

Let us substitute expression (23) for

~ ∂j eext ac 2 ~ eext IV = ( ∫ [−ϕµ ( j ) − t 2 ∂ V

and

δ

In order to deal with a function that is continuous along with its derivatives, we, for a while, will not pass

Taking into account equality (21), we obtain the

t′ = r / c ,

D0

becomes the Dirac

∫ D0 (t )dt = 1 .

~ 1 ∂ ~ ∂ϕ ∂E Σ1 = 2 ( E −ϕ ). ∂t ∂t c ∂t

1 r

function. Function

D0

−∞

The direct check shows that

ϕ = D0 =

, function

∞

~ 1 ~ ∂ 2ϕ ∂2E Σ1 = 2 ( E 2 − ϕ 2 ) . c ∂t ∂t

expression for volume integral

γ →0

As

(25)

~ 1 ~ ∂ϕ ∂E ∞ = 2 (E − ϕ ) |− ∞ = 0 . ∂ t ∂t c Let us pass to the limit in (26) as

γ →0

(27)

and use (27):

Studies in System Science (SSS) Volume 1 Issue 1, March 2013

∞

lim

γ →0

∫

IV dt =

t = −∞

~ 1 ∂j eext ac 2 ~ eext ′ µ D t t j ) [ lim ( ) ( − + − ∫ ∫ r γ →0 0 2 ∂t t = −∞V ∞

a c ~ ~ mext 1 1 1 )E − [ j − lim D0 (t + t ′)(b − × ( lim D0 r γ →0 r r γ →0 4 ∂ lim D (t + r / c) 1 γ →0 0  )r0 ] − c ∂ (t + r / c) 2 2

~ 1 − lim D0 (t + t ′)σ m j eext r γ →0 ∂ lim D (t + r / c) ~ 1 1 1 γ →0 0  )r0 divE ]dvdt + ( lim D0 − ∂ (t + r / c) r r γ →0 c

Since

(28)

~ ac 2 ~ eext 1 ∂j eext j |t = − r / c − |t = − r / c ) = ∫ [− µ ( r 2 ∂t V a 2c 2 ~ 1 1 1 ~ ) E |t = − r / c − [ j mext |t = − r / c − (b − r r r 4 ~ mext 1 ∂j  1 ~ |t = − r / c ] × r0 − σ m j eext |t = − r / c + c ∂t r

 n

is the outward normal to the surface

bounding volume

  r / r = r0 is ( x ′, y ′, z ′) .

V

and that the unit vector

directed toward observation point On surface

S1

1 r2

D0 −

, we have

gradϕ

  n = r0

.

as

1 ∂D0 (t + r / c)  )r0 , cr ∂ (t + r / c)

~   ~ (n ⋅ E ) gradϕ + [ gradϕ × [ E × n ]] 1 ∂D0 (t + r / c)   ~ 1 )r0 + = (n ⋅ E )( 2 D0 − cr ∂ (t + r / c) r ~  1 1 ∂D0 (t + r / c)  )r0 × [ E × n ]] + [( 2 D0 − cr ∂ (t + r / c) r =(

S

~  ~ ac 2 ~  H ∂ H ) × n ])ds . − − ϕ [(−σ m H − µ ( 2 ∂t

1 r2

D0 −

1 ∂D0 (t + r / c)  ~   ~  . )((n ⋅ E )n + [n × [ E × n ]]) cr ∂ (t + r / c)

~  ~   ~  (n ⋅ E )n + [n × [ E × n ]] = E , we have     (n ⋅ E ) gradϕ + [ gradϕ × [ E × n ]]  1 1 ∂D0 (t + r / c) ) = E ( 2 D0 − cr ∂ (t + r / c) r

Since

r =0 .

Following the technique from (1), we enclose this point in sphere

S1 that has small radius r1 and is centered at observation point ( x ′, y ′, z ′) thus excluding this point from region V . We have ,

Recall that

we have

~   ~ I S = ∫ ((n ⋅ E ) gradϕ + [ gradϕ × [ E × n ]]

S2

(31)

Let us consider the first surface integral and calculate the sum of the first two terms from the integrand.

gradϕ = (

Now, let us consider the surface integral from (20) and take into account replacements (13), (14), and (19):

+I

~  ~ ∂ H ac 2 ~  − ϕ [(−σ m H − µ +µ H ) × n ])ds , ∂t 2 ~   ~ I S2 = ∫ ((n ⋅ E ) gradϕ + [ gradϕ × [ E × n ]] −

Representing the expression for

~ ~ 1 1 1 ∂divE  |t =− r / c )r0 ]dvdt . (29) + ( divE |t =− r / c + r r c ∂t

S1

S1

~ ~ ∂H ac 2 ~  − ϕ [(−σ H − µ +µ H ) × n ])ds . ∂t 2

δ

IV = I

~   ~ I S = ∫ ((n ⋅ E ) gradϕ + [ gradϕ × [ E × n ]] −

m

with allowance for the properties of the Dirac function, we obtain from (28) ∞ lim ∫ IV dt γ → 0 t = −∞

has a singularity at the point

S 2 is the outer surface bounding volume V ,

S2

lim D0 (t + r / c) = δ (t + r / c) ,

ϕ

where

1

γ →0

Function

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(30)

and, hence

~  ~ 1 1 ∂D0 (t + r / c) 1 I S = ∫ ( E ( 2 D0 − ) + D0σ m [ H × n ] + 1 cr ∂ (t + r / c) r r S 1

5

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Studies in System Science (SSS) Volume 1 Issue 1, March 2013

~ ∂H ac 2 ~  1 + D0 µ[( − H ) × n ])ds . ∂t r 2

(32)

Now, let us integrate expression (32) from

t=∞

and pass to the limit as

γ →0

t = −∞

to

taking into

account properties (24) and (25) of the considered functions and the properties of the Dirac

δ -function:

~ 1 lim ∫ I S dt = ∫ ∫ lim E ( 2 D0 γ → 0 t = −∞ 1 γ →0 r S t = −∞ ∞

expression (34) from the limit as

to

t=∞

and pass to

taking into account properties (24)

and (25) of the considered functions and the properties of the Dirac

δ -function:

∞ ∞  ~ 1 lim ∫ I dt = ∫ ∫ lim((n ⋅ E )( 2 D0 γ →0 t = −∞ S 2 r t = −∞ γ →0 S 2

−

∞

γ →0

t = −∞

1 1 ∂D0 (t + r / c)  )r0 + [( 2 D0 cr ∂ (t + r / c) r

1 ∂D0 (t + r / c)  ~  1  ~ )r0 × [ E × n ]] − D0σ m [n × H ] cr ∂ (t + r / c) r ~ 2 ~ 1  ∂H ac  H )])dtds − D0 µ[n × ( − r 2 ∂t ~ 1  ~ 1  ∂E  = ∫ (( 2 (n ⋅E |t = − r / c ) + (n ⋅ |t = − r / c ))r0 ∂t cr r S

1

−

1 ∂D0 (t + r / c) 1  ~ ) − D0σ m [n × H ] cr ∂ (t + r / c) r ~ 1  ∂H ac 2 ~ − D0 µ[n × ( − H )])dtds ∂t r 2 ~ 1 ~ 1 ∂E = ∫ ( 2 E |t = − r / c + |t = − r / c ∂ cr t r S −

2

~ 1 m  ~ 1  ∂H − σ [n × H |t = − r / c ] − µ[n × ( |t = − r / c . ∂t r r ac 2 ~ − H |t = − r / c )])ds 2

~ 1 ~ 1 ∂E    + [r0 × ( 2 [ E |t = − r / c ×n ]] + [ |t = − r / c ×n ]) cr ∂t r ~ 1 m  ~ 1  ∂H − σ [n × H |t = − r / c ] − µ[n × ( |t = − r / c r r ∂t

Next, we let the radius of small sphere

−

1

r1 → 0 , to obtain

S1 go to zero,

~ lim lim ∫ I S dt = 4πE |t = 0 .

1

Consider integral

I S2

(33)

over the outer surface. This

1 1 ∂D (t + r / c)   ~ (( n ∫ ⋅ E )(− r 2 D0 + cr ∂(0t + r / c) )r0 S 2

ac 2 ~ H )])ds .

2

On surface

S2 ,

 r0

observation point, i.e., into

6

is directed toward the

S2 .

∞

t = −∞

1

t = −∞

1

t = −∞

and relationships (20), (29), (30), (33), and (35) to obtain

~ ~ 1 ∂j eext ac 2 ~ eext 4πE |t = 0 = ∫ [− µ ( |t = − r / c − j |t = − r / c ) − r 2 ∂t V

~ 1 ∂j mext  1 ~ |t = − r / c ] × r0 − σ m j eext |t = − r / c + r rc ∂t

(34) vector

∞

~ 1 a 2c 2 ~ 1 ) E |t = − r / c −[ j mext |t = − r / c 2 − (b − r 4 r

~  1 1 ∂D0 (t + r / c)  )r0 × [ E × n ]] + [(− 2 D0 + cr ∂ (t + r / c) r ~ ~ 1 1 ∂H  m  − D0σ [n × H ] − D0 µ[n × ( ∂t r r −

∞

lim lim ∫ IV dt = lim lim ∫ I dt + lim ∫ I S2 dt S γ →0 r →0 γ →0 r →0 γ →0 1

integral is expressed by formula (31) and can be represented as

I S2 =

(35)

We use the equality

∞

r1 → 0 γ → 0 t = −∞

ac 2 ~ H |t = − r / c )])ds 2

Next, we integrate

~ 1 1 ~ 1 ∂divE  + ( divE |t = − r / c + |t = − r / c )r0 ]dvdt r r c ∂t ~ 1  ~ 1  ∂E  − ∫ (( 2 (n ⋅E |t = − r / c ) + (n ⋅ |t = − r / c ))r0 ∂t cr r S 2

Studies in System Science (SSS) Volume 1 Issue 1, March 2013

~ ~ eext 1 1 ∂j eext  |τ =t − r / c 2 + |τ =t − r / c ] × r0 +[ j rc ∂t r

1 ~   + [r0 × ( 2 [ E |t = − r / c ×n ]] r

~ 1 ∂E 1   ~ + [ |t =− r / c ×n ]) − σ m [n × H |t =− r / c ] cr ∂t r

~ 1 ~ 1 1 − σ e j mext |τ =t − r / c + ( divH |τ =t − r / c r r r

~ 1  ∂H − µ[ n × ( |t = − r / c ∂t r

−

ac 2 ~ H |t = − r / c )])ds .

2

(36)

Since the observation instant t = 0 has been chosen arbitrarily, (36) can be represented as

~ ~ 1 ∂j eext 4πE |τ =t = ∫ [− µ ( |τ =t − r / c r t ∂ V 2 2 ~  ac 2 ~ eext |τ =t − r / c ) − 1 (b − a c ) E |τ =t − r / c j 2 r 4 ~ ~ mext 1 ∂j mext  1 |τ =t − r / c + −[ j |τ =t − r / c ] × r0 2 rc ∂t r

−

~ 1 ~   1  ∂H  + (n ⋅ |τ =t − r / c ))r0 + [r0 × ( 2 [ H |τ =t − r / c ×n ]] ∂t cr r

~ 1 e  ~ 1 ∂H  ] |τ =t − r / c ×n ]) + σ [n × E | + [ τ =t − r / c r cr ∂t ~ 1  ∂E ac 2 ~ (38) + ε [n × ( |τ =t − r / c − E |τ =t − r / c )])ds . ∂t 2 r

exp(−iωt )

Conclusions

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~ 1  ∂E 1 ~    |τ =t − r / c ))r0 + [r0 × ( 2 [ E |τ =t − r / c ×n ]] + (n ⋅ cr ∂t r

~ 1 ∂E 1   ~ + [ |τ =t − r / c ×n ]) − σ m [n × H | ] τ =t − r / c cr ∂t r

~ 1  ∂H − µ[ n × ( |τ =t − r / c ∂t r (37)

Applying the duality principle to (37), we obtain the integral representation for the vector of the intensity of a nonstationary magnetic field

1 a 2c 2 ~ ac 2 ~ mext ) H |τ =t − r / c − j |τ =t − r / c ) − (b − r 4 2

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e m and when σ = σ = 0 , formulas (37) and (38) become the integral representations known as the Stratton--Chu formulas [1, Section 8.14, formulas (19) and (20)].

~ 1 ∂divE  1  ~ + |τ =t − r / c )r0 ]dvdt − ∫ (( (n ⋅E |τ =t − r / c ) c ∂t r2 S

~ ~ 1 ∂j mext 4πH |τ =t = ∫ [− ε ( |τ =t − r / c r ∂t V

~ ~ 1 ∂divH  + |τ =t − r / c )r0 ]dvdt − ∫ (( 1 (n ⋅H |τ =t − r / c ) 2 c ∂t r S

When all functions depend on time as

~ 1 1 1 ~ − σ m j eext |τ =t − r / c + ( divE |τ =t − r / c r r r

ac 2 ~ − H |τ = t − r / c )])ds .

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Thus, an analog of the Kirchhoff integral representation has been obtained for the vectors of the intensities of nonstationary electric and magnetic fields in a homogeneous isotropic conducting medium. According to the dereived relationships, the field at any instant at an arbitrary point in a closed volume is expressed through an integral over the surface bounding this volume and an integral over the volume. The integrand functions contain the values of the field intensities, external currents and charges, and the time derivatives of these quantities at instants preceding the observation moment.

The obtained time-domain integral representations for nonstationary electric and magnetic fields can be used for deriving integral equations whose application will make it possible to extend the method of time-domain integral equations to problems of electrodynamic analysis of structures with conducting media. In particular, this method can be applied to simulation of diffraction of a nonstationary electromagnetic field in the case when a scatterer and (or) the medium in which it is located exhibit (exhibits) conductivity.

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Studies in System Science (SSS) Volume 1 Issue 1, March 2013

REFERENCES

Il’inskii, Anatolii S., and Slepyan, Grigorii. Ya. Oscillations and Waves in Lossy Electrodynamic Systems. Moscow.

Stratton, Julius A. Electromagnetic Theory. New York: McGraw-Hill, 1941.

Mittra, Raj, ed. Computer Techniques for Electromagnetics.

New York: Pergamon, 1973.

Vasil’ev, Evgenii N. Excitation of Bodies of Revolution.

Moscow: Radio i Svyaz', 1987) [in Russian].

Izdatel’stvo Moskovskogo Univeresiteta, 1983. [in Russian].

Felsen, Leopold B, ed. Transient Electromagnetic Fields. Vol. 10 of Topics in Applied Physics. New York: SpringerVerlag, 1977.

Colton, David, and Kress, Rainer. Integral Equation Methods in Scattering Theory. New York: Wiley, 1983.

Vasil’ev, Evgenii N., and Efimova, Irina. G. “Electric Currents on Ideally Conducting Body of Revolution during Transient Diffraction.” Radiophysics and Quantum Electronics (27) 1984: 66—73.

Efimova, Irina. G., and Sedel’nikova, Zoya V. “Diffraction of an Electromagnetic Signal on an Ideally Conducting Cone with a Dielectric Coating.” Computational Mathematics and Modeling (10) 1999: 249—53.

Efimova, Irina. G. “The Uniqueness of a Solution to the Time-Domain Integral Equation for the Nonstationary Diffraction Problem.” Journal of Communications Technology and Electronics (53) 2008: 1399—401.

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