[PDF] [PDF] Maxwells Equations in Differential Form

[PDF] Maxwells Equations in Differential Form

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Maxwell's Equations in Differential Form

dciciidi

JJJtDJJHMMMtBE

mvev BD tBM d tDJ d E

Electric field intensity [V/m]

B Magnetic flux density [Weber/m

2 = V s/m 2 = Tesla] i M

Impressed (source) magnetic current density [V/m2

d

M Magnetic displacement current density [V/m

2

H Magnetic field intensity [A/m]

i J Impressed (source) electric current density [A/m 2 D Electric flux density or electric displacement [C/m 2 c

J Electric conduction current density [A/m

2 d

J Electric displacement current density [A/m

2 ev

Electric charge volume density [C/m

3 mv

Magnetic charge volume density [Weber/m

3

Remarks:

1.

Impressed magnetic current density (

i

M) and magnetic charge density (

mv ) are unphysical quan tities introduced through "generalized" current to balance

Maxwell's equations.

2.

Although unphysical,

i M, mv similar to i Jand ev can be considered as energy sources that generate the fields. 3.

Through "equivalent principle"

i Mand mv can be used to simplify the solutions to some boundary value problems. 4. tBM d (magnetic displacement current density [V/m 2 ]) is introduced analogous to tDJ d (electric displacement current density [A/m 2 i J

Current source

c J dtDJ d i

J generates the electric

displacement current

Integral form of Maxwell's Equations

Elementary vector calculus:

Stokes' Theorem:

CS ldAsdA • It says that if you want to know what is happening in the interior of a surface bounded by a curve just go around the curve and add up the field contributions.

Divergence Theorem:

SV sdAdVA • In simple words, divergence theorem states that if you want to know what is happening within a volume of V just go around the surface S(bounding volume V) and add up the field contributions.

Null Identities:

EEABBA

atic)(electrost 000 0 • The Divergence and Stokes' theorems can be used to obtain the integral forms of the

Maxwell's Equations from their differential form.

Si SSi sdMsdBtsdEMtBE volume V surface S S C Si SC sdMsdBtldE Si SSSi sdJsdEsdtDsdHJEtDH Si Sc SC sdJsdJsdDtldH Where EJ c e Vev SVev Vev

QdvsdDdvdvDD

Vmvm

VmvVmv

dvQsdBdvdvBB Since 00 Sm sdBQ S C S V

Helmholtz Theorem

Traditionally, Newtonian mechanic is formulated in terms of force (

F) and torque (

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