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MAXWELLS EQUATIONS

Monochromatic (phasor, single frequency

sinusoidal, harmonic) form

Local (differential) form

Integral form

Electromagnetic Power and Poynting Theorem

ENERGY DENSITY

LORENTZ FORCE (Mechanical force)

CONSERVATION OF ENERGY IN EM FIELDS

Total power generated by the sources

(batteries, generators, etc.)

Rate if increase of electric and magnetic stored

energy

Power density emanating RADIATION

Power dissipated

Proof of Conservation of EM energy from the

Maxwells Equations

Quod erat demonstrandum

EM energy stored in Volume V

Dissipation in conductors

Generators produce () or dissipate (-)

energy/unit time

Power leaves the volume through RADIATION

Poynting vector

Conservation of EM energy Illustration of the

terms

The energy flows in dielectrics, not in wires

Illustration

The energy flows from the generator to the load

in the dielectric (not in the wires)

If the wires are lossy, the energy which is

dissipated by the wires flows from the field into

the wires through the surface of the wires

The energy per unit time flowing into the wire of

length l through its surface

This is the Joule power dissipated by the wire

IF IS NOT ZERO ? ENERGY

FLOWS

If we discharge the capacitor, the electric

field disappears and the circulation of energy

stops

The discharge and the magnetic field interacts,

and the momentum of the interaction is equal to

the momentum of the circulating EM energy

Mass

Momentum

UNIQUENESS OF THE SOLUTION OF MAXWELLS EQUATIONS

We assume, that there are TWO SOLUTIONS, both

satisfying the SAME INITIAL AND BOUNDARY

CONDITIONS. We show that THEY ARE IDENTICAL.

Proof

Two solutions

Both satisfies Maxwells Equations, initial and

boundary conditions. From linearity it follows

that their difference also satisfy them.

We can wrtite for the diffenrence

Generators are the same, and at the boundary

fields are the same, thus

Note that the rhs is always positive, thus the

total energy can only decrease in time

From the initial conditions it follows that at t

t0 the total energy is ZERO. The total

energy can never be negative, thus

Quod erat demonstrandum

IF inside a closed volume THE INITIAL

CONDITIONS are given , i.e. the ELECTRIC AND

MAGNETIC FIELDS ARE KNOWN at a given moment

of time t t0,

AND the GENERATORS either the TANGENTIAL

ELECTRIC or the TANGENTIAL MAGNETIC fields are

given at the SURFACE enclosing the volume from

t0 to t (BOUNDARY CONDITIONS),

THEN the ELECTROMAGNETIC FIELD INSIDE THE VOLUME

can be determined from the Maxwell equations

UNIQUELY.

Power generated inside the volume

Increases the total EM energy

Dissipated power

Radiated power

MAXWELLS EQUATIONS ? KIRCHOFFS EQUATIONS

KIRCHHOFF I. CONSERVATION OF CHARGE

KIRCHHOFF II. CONSERVATION OF ENERGY

IN DIRECT CURRENT CIRCUITS (DC)

GENERATORS AND IDEAL CIRCUIT ELEMENTS

Battery

GEBERATORS CHARGE SEPARATORS

Lineáris ido-invariáns koncentrált paraméteru

áramkörök

Ha kondenzátor is van az áramkörben, akkor

TRANSMISSION LINE AS LINEAR DISRIBUTED CIRCUIT

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