LECTURE # 15

CONDUCTOR – FREE SPACE BOUNDARY CONDITIONS

Introduction

+ + + + + + + +

- - - - - - - -

conductor

free space

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OBJECTIVES

To relate mathematically fields that propagates between various materials.

To derive boundary conditions at the interface.

GRAPHICAL ILLUSTRATION

Medium # 1

Medium # 2

E1

E2

E1t + E1n

E1n

E2t

E2n

E1 E2

E1t

E2n + E2t

Normal component

Tangential component

NORMAL COMPONENTSE1n E2n

# 1

# 2

+ + + +h

s

E1n

E2n

S Maxwell’s equation (Gauss’s law)

ensQsdD

1

2 3

321 sdDsdDsdD

s sds

zD n1

zds1 zD n2 zds3

Assume h → 0

SSDSD sn2n1

sn2n1 DD Boundary condition for normal components

0

TANGENTIAL COMPONENTS

# 1

# 2

E1t E2t

E1t

E2t

h

ℓ

1 2

34

Maxwell’s equation (Conservation of energy)

0dE

2

1

3

2

4

3

1

4dEdEdEdE

0EE t2t1 t2t1 EE

Assume again h → 0

Boundary conditionfor tangentialcomponents

0 0

CONDUCTOR – FREE SPACE BOUNDARY CONDITION# 1 Free space # 2 Conductor

0 0

Boundary condition for normal components

Boundary condition for tangential components

sn2n1 DD

t2t1 EE

sn1D 0

00EE t2t1

Boundary conditionsfor conductor – freespace/dielectric

GRAPHICAL ILLUSTRATION

Conductor Free space+

++

+

+

+

+

+

s

nsaD

Unit vectornormal to thesurface

n0

s aE

SUMMARIZED THE PRINCIPLES WHICH APPLY TO CONDUCTORS IN ELECTROSTATIC FIELDS The static electric field intensity inside a conductor is

zero. The static electric field at the surface of a conductor

is everywhere directed normal to that surface. The conductor surface is an equipotential surface.

EXAMPLE 15.1

Let potential field V = 100(x2 - y2) and point P( 2, -1, 3) lies on a conductor-free space boundary. Determine the profile of the conductor. Determine the electric field intensity at point P. Determine the surface charge at point P.

EXAMPLE 15.2

A potential field is given as V = (100e-5x sin 3y cos 4z) V. Let point P(0.1, ) be located at a conductor-free space boundary. At point P, find the magnitude of: (i) V; (ii) E; (iii) En; (iv) Et; (v) s.

Answer: 37.1 V, 233 V/m, 233 V/m, 0, 2.06 nC/m2

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