Electromagnetism INEL 4152 CH 9 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayag ü ez, PR

In summary  Stationary Charges  Steady currents  Time-varying currents  Electrostatic fields  Magnetostatic fields  Electromagnetic (waves!) Cruz-Pol, Electromagnetics UPRM

Outline  Faraday’s Law & Origin of emag  Maxwell Equations explain waves  Phasors and Time Harmonic fields Maxwell eqs for time-harmonic fields Maxwell eqs for time-harmonic fields

Faraday’s Law 9.2

Cruz-Pol, Electromagnetics UPRM Electricity => Magnetism  In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. This is what Oersted discovered accidentally:

Cruz-Pol, Electromagnetics UPRM 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!

Cruz-Pol, Electromagnetics UPRM Electromagnetics was born!  This is Faraday’s Law - the principle of motors, hydro-electric generators and transformers operation. *Mention some examples of em waves

Faraday’s Law  For N =1 and B =0 Cruz-Pol, Electromagnetics UPRM

Transformer & Motional EMF 9.3

Three ways B can vary by having… 1. A stationary loop in a t-varying B field 2. A t-varying loop area in a static B field 3. A t-varying loop area in a t-varying B field Cruz-Pol, Electromagnetics UPRM

1. Stationary loop in a time-varying B field Cruz-Pol, Electromagnetics UPRM

2. Time-varying loop area in a static B field Cruz-Pol, Electromagnetics UPRM

3. A t-varying loop area in a t-varying B field Cruz-Pol, Electromagnetics UPRM

Transformer Example Cruz-Pol, Electromagnetics UPRM

Displacement Current, J d 9.4

Cruz-Pol, Electromagnetics UPRM Maxwell noticed something was missing…  And added J d, the displacement current I S2S2 S1S1 L At low frequencies J>>J d, but at radio frequencies both terms are comparable in magnitude.

Maxwell’s Equation in Final Form 9.4

Cruz-Pol, Electromagnetics UPRM 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/m 2 ]

Cruz-Pol, Electromagnetics UPRM Maxwell Equations in General Form Differential form Integral Form Gauss’s Law for E field. Gauss’s Law for H field. Nonexistence of monopole Faraday’s Law Ampere’s Circuit Law

Cruz-Pol, Electromagnetics UPRM Maxwell’s Eqs.  Also the equation of continuity  Maxwell added the term to Ampere’s 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. This added term is called the displacement current density, while J is the conduction current.

Relations & B.C. Cruz-Pol, Electromagnetics UPRM

 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 Faraday’s law: Cruz-Pol, Electromagnetics UPRM

Electric & Magnetic potentials:  If we take the divergence of E :  We have:  Taking the curl of: & add Ampere’s we get Cruz-Pol, Electromagnetics UPRM

Electric & Magnetic potentials:  If we apply this vector identity  We end up with Cruz-Pol, Electromagnetics UPRM

Electric & Magnetic potentials:  We use the Lorentz condition: To get: and: Cruz-Pol, Electromagnetics UPRM Which are both wave equations.

 Time Harmonic Fields Phasors Review 9.7

Time Harmonic Fields  Definition: is a field that varies periodically with time. Ex. Sinusoid Ex. Sinusoid  Let’s review Phasors! Cruz-Pol, Electromagnetics UPRM

Phasors & complex #’s Working with harmonic fields is easier, but requires knowledge of phasor, let’s review  complex numbers and  phasors

Cruz-Pol, Electromagnetics UPRM COMPLEX NUMBERS:  Given a complex number z where

Cruz-Pol, Electromagnetics UPRM Review:  Addition,  Subtraction,  Multiplication,  Division,  Square Root,  Complex Conjugate

Cruz-Pol, Electromagnetics UPRM For a Time-varying phase Real and imaginary parts are:

Cruz-Pol, Electromagnetics UPRM PHASORS  For a sinusoidal current equals the real part of  The complex term which results from dropping the time factor is called the phasor current, denoted by (  The complex term which results from dropping the time factor is called the phasor current, denoted by (s comes from sinusoidal)

Cruz-Pol, Electromagnetics UPRM Advantages of phasors  Time derivative in time is equivalent to multiplying its phasor by j   Time integral is equivalent to dividing by the same term.

Cruz-Pol, Electromagnetics UPRM How to change back from Phasor to time domain The phasor is 1. multiplied by the time factor, e j  t, 2. and taken the real part.

 Time Harmonic Fields 9.7

Cruz-Pol, Electromagnetics UPRM 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.

Cruz-Pol, Electromagnetics UPRM Maxwell Equations for Harmonic fields (phasors) Differential form* Gauss’s Law for E field. Gauss’s Law for H field. No monopole Faraday’s Law Ampere’s Circuit Law * (substituting and )

Ex. Given E, find H  E = E o cos(  t-  z) a x Cruz-Pol, Electromagnetics UPRM

Ex  In free space,  Find k, J d and H using phasors and Maxwells eqs. Recall: Cruz-Pol, Electromagnetics UPRM