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ElectromagnetismINEL 4152 CH 9

- Sandra Cruz-Pol, Ph. D.
- ECE UPRM
- Mayagüez, PR

In summary

- Stationary Charges
- Steady currents
- Time-varying currents

- Electrostatic fields
- Magnetostatic fields
- Electromagnetic (waves!)

Outline

- Faradays Law Origin of emag
- Maxwell Equations explain waves
- Phasors and Time Harmonic fields
- Maxwell eqs for time-harmonic fields

Faradays Law

- 9.2

Electricity gt Magnetism

- In 1820 Oersted discovered that a steady current

produces a magnetic field while teaching a

physics class.

This is what Oersted discovered accidentally

Would magnetism would produce electricity?

- Eleven years later, and at the same time, (Mike)

Faraday in London (Joe) Henry in New York

discovered that a time-varying magnetic field

would produce an electric current!

Electromagnetics was born!

- This is Faradays Law -the principle of motors,

hydro-electric generators and transformers

operation.

Mention some examples of em waves

Faradays Law

- For N1 and B0

Transformer Motional EMF

- 9.3

Three ways B can vary by having

- A stationary loop in a t-varying B field
- A t-varying loop area in a static B field
- A t-varying loop area in a t-varying B field

1. Stationary loop in a time-varying B field

2. Time-varying loop area in a static B field

3. A t-varying loop area in a t-varying B field

Transformer Example

Displacement Current, Jd

- 9.4

Maxwell noticed something was missing

- And added Jd, the displacement current

I

L

At low frequencies JgtgtJd, but at radio

frequencies both terms are comparable in

magnitude.

Maxwells Equation in Final Form

- 9.4

Summary of Terms

- E electric field intensity V/m
- D electric field density
- H magnetic field intensity, A/m
- B magnetic field density, Teslas
- J current density A/m2

Maxwell Equations in General Form

Differential form Integral Form

Gausss Law for E field.

Gausss Law for H field. Nonexistence of monopole

Faradays Law

Amperes Circuit Law

Maxwells Eqs.

- Also the equation of continuity
- Maxwell added the term to Amperes Law so

that it not only works for static conditions but

also for time-varying situations. - This added term is called the displacement

current density, while J is the conduction

current.

Relations B.C.

?Time Varying Potentials

- 9.6

We had defined

- Electric Scalar Magnetic Vector potentials
- Related to B as
- To find out what happens for time-varying fields
- Substitute into Faradays law

Electric Magnetic potentials

- If we take the divergence of E
- We have
- Taking the curl of add

Amperes - we get

Electric Magnetic potentials

- If we apply this vector identity
- We end up with

Electric Magnetic potentials

- We use the Lorentz condition
- To get
- and

Which are both wave equations.

?Time Harmonic FieldsPhasors Review

- 9.7

Time Harmonic Fields

- Definition is a field that varies periodically

with time. - Ex. Sinusoid
- Lets review Phasors!

Phasors complex s

- Working with harmonic fields is easier, but

requires knowledge of phasor, lets review - complex numbers and
- phasors

COMPLEX NUMBERS

- Given a complex number z
- where

Review

- Addition,
- Subtraction,
- Multiplication,
- Division,
- Square Root,
- Complex Conjugate

For a Time-varying phase

- Real and imaginary parts are

PHASORS

- For a sinusoidal current
- equals the real part of

Advantages of phasors

- Time derivative in time is equivalent to

multiplying its phasor by jw - Time integral is equivalent to dividing by the

same term.

How to change back from Phasor to time domain

- The phasor is
- multiplied by the time factor, e jwt,
- and taken the real part.

?Time Harmonic Fields

- 9.7

Time-Harmonic fields (sines and cosines)

- The wave equation can be derived from Maxwell

equations, indicating that the changes in the

fields behave as a wave, called an

electromagnetic wave or field. - Since any periodic wave can be represented as a

sum of sines and cosines (using Fourier), then we

can deal only with harmonic fields to simplify

the equations.

Maxwell Equations for Harmonic fields (phasors)

Differential form

Gausss Law for E field.

Gausss Law for H field. No monopole

Faradays Law

Amperes Circuit Law

(substituting and

)

Ex. Given E, find H

- E Eo cos(wt-bz) ax

Ex. 9.23

- In free space,
- Find k, Jd and H using phasors and Maxwells eqs.

Recall