Electric and Magnetic Fields Copyright © by L.R.Linares, 2007, 2017, 2020 All rights reserved Copyright© by L.R.Linares 2007, 2017 • In this Video … • Concepts sequencing and storyboards • L.R.Linares • Scripts and Slide Production prototypes • L.R.Linares • Animation • L.R.Linares and J.C.Linares • Voice • L.R.Linares • Shooting, editing, postproduction and rendering • L.R.Linares and J.C.Linares Copyright© by L.R.Linares Goals and Expectations • Know the meaning, symbol and units for • • • • Magnetic induction (flux density) Magnetic field Magnetic potential Magnetic flux • Know the parallelism between electric circuits and magnetic circuits (cause effect chain for each) Copyright© by L.R.Linares (…) • Know the concept of magnetomotive force and how to compute it. • Numerical relationships between different units of magnetic flux, and magnetic flux density. • Know by heart the charge and mass of the electron, magnetic permeability of air. Copyright© by L.R.Linares Foundations … Copyright© by L.R.Linares Maxwell’s Equations • Everything in EE spins around Maxwell’s Equations • Everything in EE spins around Maxwell’s Equations • Everything in EE spins around Lorentz’s Forces F qE qv B So it begins … Copyright© by L.R.Linares Currents create magnetic flux Who creates what? f (Mx) (lines) (Wb) (weber) (kMx) (kilolines) i(A) i(A) 1Wb 100 MMx=10 lines 8 Copyright© by L.R.Linares Magnetic flux • In the “old days” physicists used to measure the magnetic flux in… • … lines. • They would say f=3700 lines • Or perhaps f=3.7 kilo lines • That unit, the line, is now called maxwell • So we would say f=3.7kMx Copyright© by L.R.Linares Magnetic flux (…cont…) • The modern SI unit for magnetic flux is the weber, a huge unit: • One weber = a hundred million lines. • So we end having fluxes of some milliwebers, or even microwebers: f 3.7 kMx f 37 Wb Copyright© by L.R.Linares How closely packed the flux… Some times is NOT called the magnetic Magnetic field Induction B is aBconsequence of the magnetic field • Very often, we are interested, not in how much flux we have … • … but in how closely packed together those lines are … • … the magnetic flux density … B • …B is measured in … • Wb/m2 = T (tesla) Mx/cm2 = G (gauss) Cross section of the flux Copyright© by L.R.Linares Top Hat L2.a • If one weber is a hundred million lines (maxwells), a gauss is a density of one line per square cm, and a tesla is a density of one weber per square meter, how many gauss are equivalent to a tesla of flux density? Copyright© by L.R.Linares Summary so far • Currents, amps, create flux f in webers. • Flux density, B, is measured in teslas, T. • B is not the magnetic field. • B is a consequence of the magnetic field. Copyright© by L.R.Linares Our good Old Friend The Electric Field Copyright© by L.R.Linares GUY MOVING CHARGE ELECTRIC CHARGE (C) ) ELECTRIC FIELD Copyright© by L.R.Linares Price Tag • Everything in this universe has a price tag (with one or two lonely exceptions) • So … to “move” electric charge against an electric field … we have to pay … • Pay? Pay what? And to whom? • Well, we pay Nature, if you will … through her proxy … the Electric Field … • We pay the only “currency” she accepts: ENERGY … iN joulEs Copyright© by L.R.Linares In an Electric Field ELECTRIC POTENTIAL +q J V C J /C E 30 m V E 30 m Copyright© by L.R.Linares Copyright© by L.R.Linares Units of the Electric Field V N m C Copyright© by L.R.Linares ? Units of the Electric Field V J / C N .m / C N m m m C Copyright© by L.R.Linares Volt = electric potential • To move one positive coulomb from A to B • … we spend 37 joules • We say that the difference of electric potential between A and B is 37 J/C • Or we simply say the “voltage” between A and B is 37 volts. • Or that B is 37 volts higher than A • Do remember that J/C = V Copyright© by L.R.Linares From a source… • A “battery” of 100V, applied between two points separated by 5 metres … • … creates an electric field between those points: 100 V V N E 20 20 5m m C • We call batteries and generators, electromotive forces, emf, measured in volts. Copyright© by L.R.Linares Summary so far • Currents, amps, create flux f in webers. • B is flux density, in Wb/m2, in teslas. • B is not the magnetic field. • B is a consequence of the magnetic field. • Electric fields, E in V/m … • … are created by … • … electromotive forces, emf, in volts Copyright© by L.R.Linares Our good New Friend The Magnetic Field Copyright© by L.R.Linares In a Magnetic Field +UMC Copyright© by L.R.Linares Copyright© by L.R.Linares Copyright© by L.R.Linares In a Magnetic Field MAGNETIC POTENTIAL +UMC H 30 J / UMC H 30 m Vmag MAGNETIC VOLT, Vmag m Copyright© by L.R.Linares Copyright© by L.R.Linares In a Magnetic Field MAGNETIC POTENTIAL J / cooper H 30 m +cooper J Vmag cooper Copyright© by L.R.Linares J A cooper Two jobs for the ampere Copyright© by L.R.Linares By day, the ampere is the unit of currents! Copyright© by L.R.Linares By night, the ampere is the unit for magnetic potential! … a magnetic “volt” of sorts! Copyright© by L.R.Linares So … what’s ampere? • Ampere is the unit for electric current C A s • Ampere is also the unit for magnetic potential JJ A cooper UMC Copyright© by L.R.Linares Units of E V E 30 m Copyright© by L.R.Linares Units of H A H 30 m Copyright© by L.R.Linares Every effect has a cause! Chain of cause effect for… S f (Wb) B d S Copyright© by L.R.Linares S emf (V ) E J E I J d S mmf ( A) H B H f B d S So … we apply volts (emf), and we get current? So … we apply amps (mmf), and we get flux? Copyright© by L.R.Linares V A S emf (V ) E J 2 E I ( A) J d S m m S Unit of electric potential S A Wb mmf ( A) H B 2 H f (Wb) B d S m m S Copyright© by L.R.Linares V A S emf (V ) E J 2 E I ( A) J d S m m S Unit of magnetic potential S A Wb mmf ( A) H B 2 H f (Wb) B d S m m S Copyright© by L.R.Linares V A S emf (V ) E J 2 E I ( A) J d S m m S We get amps S A Wb mmf ( A) H B 2 H f (Wb) B d S m m S Copyright© by L.R.Linares V A S emf (V ) E J 2 E I ( A) J d S m m S We get webers S A Wb mmf ( A) H B 2 H f (Wb) B d S m m S Copyright© by L.R.Linares OHM’S LAW V A S emf (V ) E J 2 E I ( A) J d S m m S OHM’S LAW S A Wb mmf ( A) H B 2 H f (Wb) B d S m m S Copyright© by L.R.Linares Magnetomotive Force MMF … in amps! Copyright© by L.R.Linares Hmm… what is an MMF? • The (CW) MMF applied to a (any) closed path is computed as follows: • Imagine a surface limited by that path • Add all the amps that (from our perspective) enter into the surface … • … subtract all the amps that (from our perspective) come out of the surface … • The resulting amps is the total CW MMF applied to that closed path! Copyright© by L.R.Linares What is the MMF (CW)? • … on this path Copyright© by L.R.Linares Imagine a surface limited • … by the path Copyright© by L.R.Linares Add current entering … • … the surface Copyright© by L.R.Linares subtract currents leaving … • … the surface MMFcw 725A Copyright© by L.R.Linares What’s the CW emf in the left loop? + _ 5V 12V + _ 2V _+ emf=5 12 2 15 V Copyright© by L.R.Linares + _ 7V What’s the CW MMF in the left window? I1 N 1 N 3 I2 I 3 N 5 N2 I 4 N 4 Copyright© by L.R.Linares I5 What’s the MMF in the left window? I1 N 1 N 3 I2 I 3 N 5 N2 I 4 N 4 Copyright© by L.R.Linares I5 What’s the MMFCW in the left window? MMFaCW N1 I1 N 2 I 2 N 4 I 4 N3 I 3 I1 NN11 I3 NN33 I2 I5 N 5 N N22 I 4 N 4 MMFCW Copyright© by L.R.Linares entering surface i i leaving surface Top Hat (L2.1, L2.2) What is the total clockwise magnetomotive force applied to the right window. I1 N 1 I2 N 3 I 3 I5 N 5 N2 I 4 N 4 N1 500 I1 3 A N 2 700 I 2 5 A N 3 1000 I 3 7 A N 4 50 I 4 200 A N 5 750 I 5 2.7 A What’s the MMFCW in the left window? MMFaCW N1 I1 N 2 I 2 N 4 I 4 N3 I 3 I1 NN11 I3 NN33 I2 N 5 N I5 N22 I 4 N 4 MMFbCW N 5 I 5 N3 I 3 Copyright© by L.R.Linares What’s the MMF in the left window? I a N1I1 N 2 I 2 N 4 I 4 N3 I 3 What is the CW-MMF applied to the left window? … to the right window? Ia Ib I b N5 I 5 N3 I 3 Copyright© by L.R.Linares Magnetic Potential Drop! (For Wednesday, 2023 Sep 21) Copyright© by L.R.Linares Top Hat.3 • If the electric field is constant, 7.5 V/m, in a segment of an electric circuit of 3.2 metres long, what is the total voltage drop from one end (A) to the other (B) of that segment, in volts? B A Copyright© by L.R.Linares Top Hat.4 • If the electric field is constant, 7.5 V/m, in the segment of an electric circuit shown, what is the total current, in amps? B A Copyright© by L.R.Linares Top Hat.5 • If the magnetic field is constant, 9.2 A/m, on a segment of an magnetic circuit of 0.25 metres long, what is the total magnetic potential drop from one end to the other of that segment, in amps? B A Copyright© by L.R.Linares Top Hat.6 • If the magnetic field is constant, 9.2 A/m, on a segment of an magnetic circuit shown, what is the flux from A to B, in Wb? B A Copyright© by L.R.Linares Top Hat.7 • If the magnetic field is constant, 200 A/m, on a segment of an magnetic circuit shown, what is the flux from A to B, in Wb? B A Copyright© by L.R.Linares Voltage drop … Amps drop? V El U H l Copyright© by L.R.Linares Electric Potential Drops • If that electric field is constant along a segment of the circuit of length l(m)… • … the electric potential drop along that segment is computed simply as V E.l Copyright© by L.R.Linares KVL • In any loop, the sum of emf, equates the voltage drops n E l emf 1 Copyright© by L.R.Linares Magnetic Potential Drops • If that magnetic field is constant along a segment of the circuit of length l(m)… • … the magnetic potential drop (amps) along that segment is computed simply as U H .l Copyright© by L.R.Linares The KVL of magnetic circuits: Ampere’s Law • In any window, the sum of mmf, equates the magnetic potential drops n H l mmf 1 m n H l N I k 1 k k 1 A window with ‘m’ coils, each one with Nk turns, and a current Ik It has ‘n’ segments with constant H. Copyright© by L.R.Linares KCL of magnetic circuits • In electric circuits, we relied on three laws to solve them: Ohm’s, KVL, and KCL. • In magnetic circuits: for any Gauss surface, the sum of all the In any Gauss surface, magneticKCL: flux going into the surface, equals thethe sumsum of all of the magnetic flux out of the surface. all coming the current going into the surface equates the sum of all the currents coming out of the surface B 0 i f fi in in Gauss Gauss out out Gauss Gauss Copyright© by L.R.Linares What about Ohm’s Law? reluctance or magnetic resistance • If we can assume that “mu” is constant • … across a segment of the magnetic circuit with a given geometry… • … we can approximate the magnetic potential drop (amps) across the segment as proportional to the flux (webers) through the segment U f amps Copyright© by L.R.Linares webers Magnetic Resistance? • If we want a flux f webers from A to B … what is the potential drop from A to B, in amps? B f A f H A f l U l f Rf A A RELUCTANCE Copyright© by L.R.Linares A word about units • The first unit for E that we learned came from Coulomb’s law, about the force that an electric field applies a charge q F qE • So the unit for E were given as N/C • Yes, force per unit of charge. • Now we have seen it can also be V/m • But, H is given in A/m (mag. Pot. Per metre) Copyright© by L.R.Linares A word about H units • Originally, physicists in Europe used the cgs system (not the metric, SI, system). • In cgs, the force unit was the dyne (10-5N) • The unit of magnetic charge was the “unit-pole”, so the magnetic field H was given in dyne 1 Oe 1 unitpole • 1 oersted equals (1000/4p) A/m Copyright© by L.R.Linares Unit pole • The concept of the unit pole as an integral part of the presentation of magnetic fields was present in Physics’ books until the 1930s, and even later for some authors. Copyright© by L.R.Linares Flux Density, B • What creates the flux density is H, the magnetic field. • The same way that what creates the current density J is E, the electric field. • B and H are related, but are not the same. • B depends on H, but also on the magnetic permeability of the material or medium. Copyright© by L.R.Linares THE END Copyright© by L.R.Linares Caboose Copyright© by L.R.Linares Copyright© by L.R.Linares Copyright© by L.R.Linares So … the number of joules per metre and per kilogram isGravitational a measure Field of how strong the gravitational field is The eagle is pulling up the fish! J / kg G 30 m For every metre the eagle raises 1kg of fish, the eagle spends “so many” joules. On Jupiter it would Need to spend more Joules per metre per Kilogram! AGAINST THE FIELD! Copyright© by L.R.Linares Units of the Electric Field J /C N m C Copyright© by L.R.Linares B, Wb/m2 H, A/m Copyright© by L.R.Linares Ridiculosity is forever! Copyright© by L.R.Linares Ridiculosity is forever! Copyright© by L.R.Linares B, flux density! LOW B hairs 2 cm Copyright© by L.R.Linares B, flux density! HIGH B hairs 2 cm Copyright© by L.R.Linares Example, a cathode ray tube Copyright© by L.R.Linares 10000 V V N E 100, 000 100, 000 0.1 m m C The force on an electron: - 10,000V + - - 10cm F qE N 1.6022 10 C 100, 000 C 14 19 1.602 10 N The acceleration on an electron: F 1.602 1014 N 15 m F ma a 17.6 10 2 31 m 9.1110 kg s The time to travel the cannon: 2 at d vot Yes, 302, not 289, not 2the standard! m 17.6 10015to t 2 100km/h in 6s) aMustang 4.63 2 (3020.1 cu.in. V8 t 3.37ns s 2 Copyright© by L.R.Linares TOP HAT FLD.1 What is the time it takes an electron to travel between the plate on the left and the plate on the right, in nano seconds? - 10,000V + - - 10cm Copyright© by L.R.Linares In a Magnetic Field MAGNETIC POTENTIAL J / UMC H 30 m +UMC J A UMC A H 30 m Copyright© by L.R.Linares emf (V ) E (V / m) J ( A / m ) E 2 S I ( A) J d S Every effect has a cause! S Unit of electric Chain potential of cause effect for… To current… From voltage… mmf ( A) H ( A / m) B(Wb / m ) H 2 S f (Wb) B d S Copyright© by L.R.Linares S In a Magnetic Field MAGNETIC POTENTIAL J / cooper H 30 m +cooper J Vmag cooper Copyright© by L.R.Linares A H 30 m