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MAXWELLS

EQUATIONSAND BOUNDAR

CONDITIONS

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INTRODUCTION

Maxwells equations characterize macroscopic

matter by means of its permittivity , permeability

, and conductivity , where these properties are

usually represented by scalars and can vary amongmedia.

If the field exists in a region consisting of two

different media, the conditions that the field must

satisfy at the interface separating the media are

called boundary conditions.

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E Electric fieldH Magnetic field

D Electric displacement

B Magnetic induction

r Electric charge density

J Electric current density

e Permittivity

m Permeability

e 0 Permittivity of vacuum

m0 Permeability of vacuum

P Electric polarization

M Magnetization

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Boundary Conditions

for agnetic Field

The derivation of boundary conditions for the

magnetic field, follows similar arguments to

that of the electric field, but using equations:

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Again we consider the normal and

tangential components as illustrated

below.

Normal and tangential components the B field on eitherside of an interface.

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Normal Component of B

The boundary condition for the normal

component of the magnetic field can be

obtained by applying Gausss flux law:

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Now if two planes are allowed to approach one

another, keeping the interface between them, the

area of the curved surface will approach zero,

giving:

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Tangential Component of H

We can derive the tangential component of H

by applying Amperes law

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The variation in H across an interface is

obtained by the application of this law around a

closed rectangular path, as shown in the figure.Assuming no current at the interface, letting the

rectangle shrink to zero:

Thus, tangential component of H has the same

projection along the two sides of the rectangle.

Since the rectangle can be rotated 90 and theargument is repeated, it follows.

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The tangential component of H is continuous across a

current-free interface. The relation

between the angles made byH1

and H2

with a current-

free interface (see figure below) is obtained.

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Current Sheet at

the Boundary

The magnetic field H produced by

this current sheet is given by:

The resultant magnetic field,

This is expressed by the vector

formula

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For reference purposes, the relationship for E and

D across the interface of two dielectrics are shown

below along the relationships for H and B.

agnetic Fields

B

n1= B

n2

H

t1= H

t2(Current-free)

(H1 H2) an12 = K (with current sheet)

(current-free)

Summary of

Boundary Conditions

Electric Fields

D

n1= D

n2(Charge-free)

(D1 D2) an12 = -r (with surface charge)

t1=

t2

(charge-free)

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axwell's equationsare a series of four partial

differential equations that describe the force of

electromagnetism. Individually, the four equations are

named Gauss' law, Gauss' law for magnetism, Faraday's

law and Ampere's law.

General form of axwells equations:

1. It actually applies to all cases, either in vacuum or in amedium. It is thus called the general form of axwells

equations.

2. It is also called axwells equations in vacuum.

AXWELLS

EQUATIONS

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Maxwells Equations, General Set

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Maxwells Equations, Free-space Set

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Thanks!Any questions?

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