Electromagnetic Boundary Condition

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4 Electromagnetic Boundary Conditions

The integral forms of Maxwells equations describe the behaviour of elec-tromagnetic field quantities in all geometric configurations. The differentialforms of Maxwells equations are only valid in regions where the parametersof the media are constant or vary smoothly i.e. in regions where (x,y,z,t),

(x,y,z,t) and (x,y,z,t) do not change abruptly. In order for a differen-tial form to exist, the partial derivatives must exist, and this requirementbreaks down at the boundaries between different materials.

For the special case of points along boundaries, we must derive the rela-tionship between field quantities immediately on either side of the boundaryfrom the integral forms (as was done for the differential forms under differ-entiable conditions).

Later, we shall apply these boundary conditions to examine the behaviourof EM waves at interfaces between different materials - from them we canderive the laws of reflection and refraction (Snells law).

4.1 Boundary Conditions for the Electric Field

Consider how the electric field E may change on either side of a boundarybetween two different media as illustrated below.

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Medium 2

Medium 1

Et2

Et1

E1

E2En2

En1

Figure 4.1: Normal and tangential components of the electric field on either side of theinterface between two media.

The vector E1 refers to the electric field in medium 1, and E2 in medium 2.One can further decompose vectors E1 and E2 into normal (perpendicular tointerface) and tangential (in the plane of the interface) components. Thesecomponents labelled En1, Et1 and En2, Et2 lie in the plane of vectorsE1 andE2.

To derive boundary conditions for E, we must examine two of Maxwellsequations:

E dl = S

B

t dS

and D dS =

V

dV

which will allow us to relate the tangential and normal components ofE oneither side of the boundary. Note that refers to the free, unbound changewithin V, i.e. it excludes the charge bound to atoms.

Normal Component ofD

The boundary condition for the normal component of the electric field canbe obtained by applying Gausss flux law

D dS =

V

dV

to a small pill-box, positioned such that the boundary sits between itsupper and lower surfaces as shown in the illustration.

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0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 01 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1

Surface charge

Medium 2

Medium 1

s

2, 2, 2

1, 1, l dS = ns

dS = ns

h

n

s

Figure 4.2: Gaussian pill box straddling the interface between two media.

If we shrink the side wall h to zero (keeping the interface sandwichedbetween the upper and lower surface) then all electric flux enters or leavesthe pill-box through the top and bottom surfaces, and

D dS D1 ns + D2 (n) s = Dn1s Dn2swhere Dn1 and Dn2 are the normal components of the flux density vectorimmediately on either side of the boundary in mediums 1 and 2, and s isthe elemental surface area.

The amount of charge enclosed as h 0 depends on whether thereexists a layer of charge on the surface (i.e. an infinitesimally thin layer ofcharge)1. If a surface charge layer exists then

V

dV = ss

and thusDn1s Dn2s = ss

1In perfect conductors, any excess free charge always resides on the surface of the conductor and isdenoted by s in units of Cm

2. Within the conductor, the charge density very rapidly goes to zero -this is discussed in a later section on relaxation time.

It should also be noted that in the case of dielectrics subjected to an electric field, the material

polarizes, which does in fact result in a surface charge layer - however this charged layer is boundcharge caused by the polarization effect, and is not part of the quantity s.

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from which we concludeDn1 Dn2 = s

For the case where s = 0,Dn1 = Dn2

Or in terms of the electric field E,

1En1 = 2En2

orEn1 =

2

1En2

Tangential Component ofE

We can derive the tangential component ofE by applying Faradays law toa small rectangular loop positioned across the boundary, and in the planeofE1 and E2, as illustrated in the diagram below.

a

cd

b

Medium 2

Medium 1

2, 2, 2

1, 1, lh

ln

E1

E2

Figure 4.3: To determine the boundary condition on the tangential component of the Efield, Faradays law is applied to rectangular loop straddling the interface be-tween two media.

Consider the limiting case where the sides h perpendicular to boundaryare allowed shrink to zero. In the limit as h 0 , the magnetic fluxthreading the loop shrinks to zero, and thus

E dl ba

E1 dl+ dc

E2 dl = 0

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E1 l+ E2 (l) = 0

Writing the tangential components ofE1 and E2 along the contour as Et1and Et2, we have

Et1l Et2l = 0

from which we conclude that on either side of the boundary,

Et1 Et2 = 0

orEt1 = Et2

i.e. tangential components immediately on either side of a boundary areequal.

4.2 Boundary Conditions for the Magnetic Field

The derivation of boundary conditions for the magnetic field, follows similararguments to that of the electric field, but using equations

B dS = 0H dl =

S

J dS+

S

D

t dS

Again we consider the normal and tangential components as illustrated be-low.

Medium 2

Medium 1Bn1B1

Bt1

Bt2

Bn2

B2

Figure 4.4: Normal and tangential components the B field on either side of an interface.

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Normal Component ofB

The boundary condition for the normal component of the magnetic field

can be obtained by applying Gausss flux lawB dS = 0

to a small pill-box.If we shrink the side wall h to zero, all magnetic flux and leaves/entersthe pill-box through the top two surfaces,

B dS B1 ns + B2 (n) s = Bn1s Bn2s

Equating to zero, we findBn1 Bn2 = 0

and hence the normal component ofB is continuous at boundaries.

Tangential Component of H We can derive the tangential component ofHby applying Amperes law to a closed loop as illustrated below. Again, therectangular loop is in the plane of vectors H1 and H2.

a

cd

b

Medium 2

Medium 1

2, 2, 2

1, 1, lh

l nH1

H2

Figure 4.5: To determine the boundary condition on the tangential component of the Hfield, Amperes law is applied to rectangular loop straddling the interface be-tween two media.

Amperes law states

H dl = S

J dS+ S

Dt

dS

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Consider the limiting case where the sides h perpendicular to the boundaryare allowed shrink to zero. The left hand side becomes

H dl b

a H1 dl+

d

c H2 dl

= H1 l+ H2 (l)

On the right hand side, the displacement current term Id =S

Dt

dS shrinksto zero. For physical media, the conductivity is finite, and J is also finite.Thus within the loop Ic =

SJ dS also shrinks to zero, and so

H1 l+ H2 (l) = 0

which implies the tangential component ofH does not change immediatelyon either side of the boundary, i.e.

Ht1 = Ht2

For the special case of an idealised perfect conductor, , a surfacecurrent may exist (i.e. current flowing within a vanishingly thin layer onthe surface). Some physical situations involving good conductors like metals

(e.g. skin effect and reflection of EM waves off metallic objects) may allow usto treat currents concentrated on the surface as a surface current modelledby a vector Js in units of Amps/m (NB not m

2) flowing in an infinitesimallythin layer. Js flows perpendicular to our rectangular loop (chosen in theplane of vectors H1 and H2), and thus

Ic =

S

J dS Jsl

where Js is the magnitude of the surface current density.We conclude that for the case where a surface current exists, the boundarycondition on the tangential component ofH is therefore

Ht1 Ht2 = Js

Summary of Boundary Conditions

Below are depicted the components on either side of the boundary in sideview.

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Medium 2

Medium 1

2, 2, 2

1, 1, lEt2

En2

Et1

En1

Bt2

Bt1

Bn2

Bn1E1

E2B2

B1

Figure 4.6: Normal and tangential components illustrated for the cases of the E field and

the B field.

The boundary conditions are summarised below.

Dn1 Dn2 = s

Et1 Et2 = 0

Bn1 Bn2 = 0

Ht1 Ht2 = Js

The boundary conditions can be expressed in vector form2 as:

D1 n D2 n = s

n E1 n E2 = 0

B1 n B2 n = 0

n (H1 H2) = Js

2These vector forms require careful 3-D visualization.

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These general conditions can be further refined depending on the specificmedia on either side of the interface. Some examples are given below.

4.3 Examples of Boundary Conditions

4.3.1 e.g. Dielectric - Dielectric Interface

Dielectrics are materials for which all electrons are bound to atoms, and arenon-conducting, i.e 0; no currents flow, and no unbound surface chargeexists unless explicitly put there (i.e. s = 0). Thus we have

Dn1 = Dn2 or 1En1 = 2En2

Et1 = Et2

Bn1 = Bn2

Ht1 = Ht2 or1

1Bt1 =

1

2Bt2

An example of a dielectric-dielectric interface is the interface between

air and glass. The above boundary conditions are applied when analysing the reflec-

tion and refraction of plane waves (studied in a later section).

For example, consider an air-glass interface where 1(air) = 0 and 2(air) =50. Given the E1 vector in the air, how can we sketch the E2 vector?

Medium 2

GLASS

Medium 1AIR

1 = 0

E1

2 = 50, 2

E2 =?

Figure 4.7: E1 field at the the interface between two dielectrics.

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We already know the tangential component must be sketched as Et2 = Et1.The normal component is related by En2 =

12En1 =

15En1. The E2 can thus

be sketched as shown below:

Medium 2

GLASS

Medium 1

AIR

1 = 0

E1

E2

En1

2 = 50, 2

En2Et2 = Et1

Figure 4.8: E1 and E2 field at the the interface between two dielectrics.

4.3.2 e.g. Dielectric - Perfect Conductor

If one of the media is a dielectric (say medium 1 is air), and the othermedium (medium 2) is a perfect conductor 2 , then En2 = 0 andEt2 = 0 inside the perfect conductor

3.

3The conductor is assumed to be stationary within the xyz frame of reference. If a conducting metalobject moves through a magnetic field, a non-zero E field can exist within the conductor. For example,if a long thin metal rod moves through a uniform B field with velocity v, the electrons inside the rodexperience a force qvB, perpendicular to the direction of motion and perpendicular to B. Electronsdeplete on the one side and increase on the opposite side, causing an induced dipole.

through uniform

magnetic field

Rod moving

Fields inside

the rod

E

v Bv

B

As the dipole forms, an electric field builds up until the internal forces balance, i.e. qE = qv B,and the electron current no longer flows. The internal field strength is E = v B. If the rod isorientated in the direction ofv B, then the potential difference between the ends of the moving rod

is =

l

0E dl = vBl where l is the length of the rod. In cases where a metal object is stationary

within the frame of reference, the electrons will rearrange rapidly (if placed in an EM field) such that

the internal electric field goes to zero; the potential difference between any two points on the conductorwill also then be zero.

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Since Dn1 Dn2 = s, we conclude that Dn1 = s

Since Et1 = Et2 and Et2 = 0, we conclude that Et1 = 0, i.e. there existsno tangential component on the dielectric side of the interface.

In vector form we state the boundary conditions for the field in the dielectricas

D1 n = s

andn D1 = 0

The E field lines always meet a perfect conductor perpendicular to the

surface, and magnetic field lines parallel to the surface as is illustratedin the figure below:

Medium 2

Medium 1

(AC fields)

Perfect conductor

dielectric e.g. air

Bn1 = 0

2, 2, 2 = inf

E1

B2 = 0

B 1, 1, lEt1 = 0

E2 = 0

En1 =

n H1 = Js

Figure 4.9: E field and the B field at the interface between air and a perfect conductor.

For AC fields, no time-varying magnetic field exists in a perfect conductor

- why? Recall that E = Bt and since E = 0 in a perfect conductor, E = 0 and hence B

t= 0. In other words, no changing magnetic field

can exist in a perfect conductor, and hence Bn2 = Bn1 = 0, i.e. the normalcomponent of the magnetic field is zero. A surface current can still exist,implying a tangential component ofB1 can exist. These two conditions canbe expressed in vector form as

B1 n = 0

n H1 = Js

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These boundary conditions are useful for establishing, for example, thecharge density or current distribution on the surface of a conductor,when the field quantifies in the dielectric are known or specified.

These boundary conditions will be applied when analysing the reflec-tion of an electromagnetic plane wave off the surface of a perfect con-ductor.

4.3.3 e.g. Conductor-Conductor (steady state current)

For the case of two conductors under static field conditions (i.e. Et

= 0 andB

t = 0), there can be no charge build up at the interface and henceJn1 = Jn2

Since Jn = En, we have an additional constraint on the normal componentof the electric field, i.e.

1En1 = 2En2

For non-steady state conditions a more complicated boundary constraint

relates J1 and J2, which can be derived by application of the continuity ofcharge equation

SJ dS =

t

VdV at the boundary. The result is

(J1 J2) n+ t Js = s

t

where t is the two-variable divergence in the tangent plane applied to thesurface current Js, and

st

is the rate of change of surface charge density inCm2s1. See [Griffiths] for details.

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Medium 2

Medium 1

Side View

surface charge

2, 2, 2

1, 1, l

Jn1 = J1 n

Jn2 = J2 n

n

s

Figure 4.10: Boundary condition for the normal component of J for conductors (for thenon-steady state case for which there may be charge build-up at the interface).

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Electromagnetic Boundary Condition