Electromagnetism INEL 4151 Sandra CruzPol Ph D ECE

Electromagnetism INEL 4151 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayagüez, PR

In summary Ø Stationary Charges l Q Ø Steady currents l E Ø Magnetostatic fields H I Ø Time-varying currents l Ø Electrostatic fields I(t) Ø Electromagnetic (waves!) l E(t) & H(t) Cruz-Pol, Electromagnetics UPRM

Outline Faraday’s Law & Origin of Electromagnetics Ø Transformer and Motional EMF Ø Displacement Current & Maxwell Equations Ø Review: Phasors and Time Harmonic fields Ø

9. 2 FARADAY’S LAW

Electricity => Magnetism Ø 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? Ø 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

Len’s Law = (-) Ø If N=1 (1 loop) Ø The time change can refer to B or S Cruz-Pol, Electromagnetics UPRM

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

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

Example PE 9. 3 A magnetic core of uniform cross-section 4 cm 2 is connected to a 120 V, 60 Hz generator. Calculate the induced emf V 2 in the secondary coil. N 1= 500, N 2=300 Ø Use Faraday’s Law Answer; 72 cos(120 pt) V

9. 3 TRANSFORMER & MOTIONAL EMF

Two cases of Ø B changes Ø S (area) changes Stoke’s theorem Cruz-Pol, Electromagnetics UPRM

Three (3) cases: Ø Stationary loop in time-varying B field Ø Moving loop in static B field Ø Moving loop in time-varying B field Cruz-Pol, Electromagnetics UPRM

Example + V 1 __ R 1=300 W + V 2 _ R 2=200 y S= 0. 5 m 2 x The resistors are in parallel, but V 2≠V 1

PE 9. 1

Vemf variation with S Ø https: //www. youtub e. com/watch? v=i-j 1 j 2 g. D 28&feature=r elated Cruz-Pol, Electromagnetics UPRM

Transformer Example Ø Find reluctance and use Faraday’s Law Cruz-Pol, Electromagnetics UPRM

9. 4 DISPLACEMENT CURRENT, JD

Ø Maxwell noticed something was missing… And added Jd, the displacement current S 1 I L S 2 At low frequencies J>>Jd, but at radio frequencies both Cruz-Pol, Electromagnetics terms are comparable in magnitude. UPRM

9. 4 MAXWELL’S EQUATION IN FINAL FORM

Summary of Terms Ø E = electric field intensity [V/m] Ø D = electric field density [C/m 2] Ø H = magnetic field intensity, [A/m] Ø B = magnetic field density, [Teslas] Ø 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

Maxwell’s Eqs. Ø Also the equation of continuity Ø Maxwell added the term to Ampere’s Law so that it not only works for static conditions but also for time-varying situations. l This added term is called the displacement current density, while J is the conduction current. Cruz-Pol, Electromagnetics UPRM

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

9. 6 TIME VARYING POTENTIALS

We had defined Ø Electric & Magnetic potentials: Ø Related to B as: Substituting into Faraday’s law: Identity: the curl of the gradient of a scalar = zero. . Choose V

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

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

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

Who was Nikola. Tesla? Ø Find out what inventions he made Ø His relation to Thomas Edison Ø Why is he not well know?

9. 7 TIME HARMONIC FIELDS PHASORS REVIEW

Time Harmonic Fields Ø Definition: is a field that varies periodically with time. Example: A sinusoid Ø 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 Ø complex numbers and Ø phasors Cruz-Pol, Electromagnetics UPRM

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

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

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

PHASORS Ø For a sinusoidal current equals the real part of Ø 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

To change back to time domain The phasor is 1. multiplied by the time factor, e jwt, 2. and taken the real part. Cruz-Pol, Electromagnetics UPRM

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

9. 7 TIME HARMONIC FIELDS

Time-Harmonic fields (sines and cosines) Ø 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. Ø 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 Cruz-Pol, Electromagnetics ) UPRM and

Example Use Maxwell equations: Ø In Phasor form Ø In time-domain Cruz-Pol, Electromagnetics UPRM

Earth Magnetic Field Declination from 1590 to 1990 Cruz-Pol, Electromagnetics UPRM