cem4042 cap 9 - ecuaciones de maxwell2014

8/10/2019 CEM4042 Cap 9 - Ecuaciones de Maxwell2014

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CA 4042: Campos

Electromagnticos

Instructor: Ing. Hctor C. Vergara V.

Profesor de Facultad de Ingeniera Mecnica

Centro Regional de Azuero

Universidad Tecnolgica de Panam

Mvil: (507) 6677-5920, email: [email protected]

Libro de Texto:M.N.O. Sadiku,Elementos de Electromagnetismo 5th ed. Oxford University Press, 2009.

Lectura Auxiliar:

W.Hayt, J.Buck, Teora Electromagntica, 8va ed. McGrawHill, 2012.

Todas las figuras son tomadas del libro de texto principal a menos que se diga lo contrario

Cap. 9:Ecuaciones de Maxwell


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Chapter 9: Maxwells Equations

Topics Covered Faradays Law

Transformer and Motional

Electromotive Forces

Displacement Current

Magnetization in Materials Maxwells Equations in Final

Form

Time Varying Potentials

(Optional)

Time Harmonic Fields (Optional)

Homework: 3, 7, 9, 12, 13, 16,

18, 21, 22, 30, 33

All figures taken from primary textbook unless otherwise cited.


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Faradays Law (1)

We have introduced several methods of examining magnetic fields in terms of forces,energy, and inductances.

Magnetic fields appear to be a direct result of charge moving through a system and

demonstrate extremely similar field solutions for multipoles, and boundary condition

problems.

So is it not logical to attempt to model a magnetic field in terms of an electric one? This is

the question asked by Michael Faraday and Joseph Henry in 1831. The result is Faradays

Law for induced emf

Induced electromotive force (emf) (in volts) in any closed circuit is equal to the time rate of

change of magnetic flux by the circuit

where, as before, is the flux linkage, is the magnetic flux, N is the number of turns in the

inductor, and t represents a time interval. The negative sign shows that the induced voltageacts to oppose the flux producing it.

The statement in blue above is known as Lenzs Law: the induced voltage acts to oppose the

flux producing it.

Examples of emf generated electric fields: electric generators, batteries, thermocouples, fuel

cells, photovoltaic cells, transformers.

dt dt

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d d NVemf


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Faradays Law (2)

The total emf generated in the between the two open terminals in the battery is therefore

Note the following important factsAn electrostatic field cannot maintain a steady current in a close circuit since

Ee dl 0IRL

An emf-produced field is nonconservative

Except in electrostatics, voltage and potential differences are usually not equivalent

Ef dl Ee dl IRVemfP P

N N

To elaborate on emf, lets consider a battery circuit.The electrochemical action within the battery results and in emf produced electric field,Ef

Acuminated charges at the terminals provide an electrostatic fieldEe that also exist that

counteracts the emf generated potential

EEf Ee

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Edl

E

fdl 0

E

f dl

P

L L N


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Electromotive Forces (1)

For a single circuit of 1 turn

The variation of flux with time may be caused by three ways

1.

2.

3.

Having a stationary loop in a time-varyingB field

Having a time-varying loop in a static B field Having

a time-varying loop in a time-varying B field

A stationary loop in a time-varyingB field

E dl dtB dS

d ddt dt

L S

Vemf

emf

d

V

V Edl E dS d B dS

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E dB

dt

dtSSLemf

One of Maxwells for time varying fields


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A time-varying loop in a static B field

Qu B

u B dl

E dS u Bdl

E u B

by_Stokes's_Theorem

Fm IlB

uBlVemf

Fm IlB

V Edl

E Fm u BQ

fieldin a motional E

F

m

L L

m

LLemf

m

Some care must be used when applying this equation

1. The integral of presented is zero in the portion of the

loop where u=0. Thus dl is taken along the portion of

the lop that is cutting the field where u is not equal

to zero

2. The direction of the induced field is the same as that

of Em. The limits of the integral are selected in the

direction opposite of the induced current, thereby

satisfying Lenzs Law



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A time-varying loop in a time-varyingB field

u B dl

dB u Bdt

E

E dl dB dSdt

V

m

LL S

emf

One of Maxwells for time varying fields

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Electromotive Forces: Example1

Conducting element is stationary and themagnetic field varies with time

Assume the bar is held stationary at y =0.08 m

andB = 4cos(106t)az mWb/m2

Assume the length between the two conducting

rails the bar slides along is 0.06 m


dt dB

E

V dB dSdt

m

S

emf

(4)(103)(106)sin(106t)dxdyS

xy(4)(103)(106)sin(106t)

0.080.06(4)(103

)(106

)sin(106

t) 19.2sin(106t)V

d (0.004cos(106t))a dSS dt

V dB dS dt

z

S

emf


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Electromotive Forces: Example

2

Conductor moves at a velocity u = 20ay m/s in

constant magnetic field B=4az mWb/m2

Assume the length between the two

conducting rails the bar slides along is 0.06 m

u B dl

E u B

V 20a 0.004a dxaL

0.08dx 0.08x 0.080.064.8mV

V E dl

xy zemf

m

L L

emf


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Conductor moves at a velocity u = 20a

ym/s in time

varying magnetic field B=4cos(106t-y)az mWb/m2

Assume the length between the two conducting

rails the bar slides along is 0.06 m


Electromotive Forces: Example 3

Edl dS u Bdld

V (103)(4)cos(106ty)a dxdya

20a (103)(4)cos(106ty)a dxaL

y z x

z

S

emf z

L S LVemf

dt

dt

dB

V (103)(4)106 sin(106ty)a dxdyaS

20(103)(4)cos(106ty)dxV (103)(4)cos(106ty)x 103(4)cos(106t)xemf

20(103

)(4)cos(106

ty)dxV (103)(4)cos(106ty)x 103(4)cos(106t)xemf 20(103)(4)cos(106 ty)xV (103)(4) 8(102 )cos(106ty)x 103 (4)cos(106 t)x

V 240cos(106 ty) 240cos(106t)

4000xcos(106ty) 4000xcos(106t)Vemf

emf

emf

zzemf


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Lets now examine time dependent fields from the perspective onAmperes Law.

H0 J

H0 J J J J v D D

t HJD

tDJ

t

We can now define the displacement current density as

the time derivative of the displacement vector

t t

HJJd

t J v 0

HJ

d

d

d

Another of Maxwells for time varying fields

This one relates Magnetic Field Intensity to conduction

and displacement current densities

Displacement Current (1)

This vector identity for the cross product is mathematically

valid. However, it requires that the continuity eqn. equals

zero, which is not valid from an electrostatics standpoint!

Thus, lets add an additional current density termto balance the electrostatic field requirement

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Using our understanding of conduction and displacement current density. Lets test this

theory on the simple case of a capacitive element in a simple electronic circuit.

H dl J dS D dS dQ IdttS

H dl J dSIenc 0

H dl J dSIenc I

DI J dS dSt

tHJD

S

dL

S2L

S1

L

d

22

If J =0 on the second surface then Jd must be

generated on the second surface to create a time

displaced current equal to current on surface 1


Based on the equation for displacement current density, we can

define the displacement current in a circuit as shown

Amperes circuit law to a closed path provides the following eqn.

for current on the first side of the capacitive element

However surface 2 is the opposite side of the capacitor and has no

conduction current allowing for no enclosed current at surface 2

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13/18

Show that Ienc

on surface 1 and dQ/dt on surface 2of the capacitor are both equal to C(dV/dt)

dQ Sds SdD SdES dV CdVdt dt dt dt d dt dt

I

d dt dt

from surface 1

SS dV CdVI Jt d dt

D dVJ

D E Vd

c

d d

d


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Maxwells Time Dependent Equations

It was James Clark Maxwell that put all of this together and reduced electromagnetic field

theory to 4 simple equations. It was only through this clarification that the discovery of

electromagnetic waves were discovered and the theory of light was developed.

The equations Maxwell is credited with to completely describe any electromagnetic field

(either statically or dynamically) are written as:

Differential Form Integral Form Remarks

Gausss Law

Nonexistence of the

Magnetic Monopole

Faradays Law

Amperes Circuit Law

t

HJD

tE B

B0

D v

B dS0S

Hdl J t dSD

SL

Edl B dStL S

D dS vdv

S


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A few other key equations that are routinely used are listed over the next couple of slides

Maxwells Time Dependent Equations (2)

E E a 0H H a KD D a B B a 02 1 n

2 n s1

2 n1

1 2 n

Boundary Conditions

Compatibility Equations

Boundary Conditions for Perfect Conductor

Equilibrium Equations

E 0

Jm

B m

Bt

E

HJD

t

Dv

m = free magneticdensity

H 0 J0

Bn 0

Et 0

Lorentz Force Law Continuity Equation

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t J v

F QE u B


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Maxwells Time Dependent Equations:

Identity Map

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Time Varying Potentials

A A 2Ayields:

A 2A J

t2

2

2V V v

Apply Lorentz Condition for potentials: A 02

t A V

by choo sing:conditionsvector field Limit the

t2V At

identity:Applying the vector

t2V At

A J

t A JE J VA

tt

dt H1 B 1 A J dD

Applying_Ampere's Circuit Law :

2

2

B t

At

E v 2V

At

E V

VAt E

t A E

At

E

Applying _Faraday 's_Law :

Definition

B A

of B from A :

Jdv4R

A

v4R

dvV

Field _potentials :

v

v

0

2

A A

Jt28/17/2012 17


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Wave Equation

0 0

0 0

t2

t2

2B 2B

2E 2E

t2

t2

2A 2A

2V

2

V

1

1

J

n c

u

c

u

y ie ld sI n fr ee s p a ce

v

Refractive index

Speed of the wave in a medium

Speed of light in a vacuum

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cem4042 cap 9 - ecuaciones de maxwell2014

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