Lesson 12 Maxwells’ Equations Gauss’ Law Faradays’ Law Amperes’ Law Ampere - Maxwell Law Maxwells Equations Integral Form Differential Form 1
ò ò Gauss' Law Gauss’ Law For Electric Fields: Q E · d A = = F e surface enclosing electric charge Gauss’ Law For Magnetic Fields: ò B · d A = = F B surface enclosing magnetic charge 2
Amperes and Faradays Laws Amperes Law ò Amperes and Faradays Laws B · d s = m I path enclosing current I B is due to I Faradays Law ò d F E · d s = - B dt path enclosing changing magnetic flux E is due to changing Flux 3
ò Faradays Law Faradays Law d F E · d s = dt - - B total dt path enclosing changing magnetic flux Change of emf around closed loop due to static Electric Field Change of emf around closed loop due to induced Electric Field 4
Changing Magnetic Flux Produces Induced Electric Field Changing Flux I 5
Maxwells Law of Induction From Gauss's law at each instant of time Maxwells Law of Induction ( ) Q t E · d A = net e ( ) If Q t is changing with time net dQ d F I = net = e E d dt dt Using Amperes ' Law we get a magnetic field given by d F B · d s = m I = m e E d d dt path enclosing changing electric flux This relationship is called Maxwells Law of Induction 6
Changing Electric Flux Produces Induced Magnetic Field Changing Flux II 7
We can thus generalize Amperes Law to look exactly analogous to Faradays’ Law Ampere - Maxwell Law I d F B · = m + m = m + m e d s I I I E tot s d s dt path enclosing changing flux and constant current Ampere - Maxwell Law 8
Displacement Current II Get varying electric fields in capacitors Ic(t) + - E(t) 10
Displacement Current I Changing electric flux produces a virtual Displacement Current d F = e I E d dt 9
Displacement Current III ( ) ( ) t Q t Displacement Current III ( ) E t = = e A e ( ) ( ) Q t Q t ( ( F ) ) t = AE t = A = E A e e F d d 1 dQ F dQ \ E = Û e E = e dt dt dt dt Þ ( ) = ( ) I t I t d c ( ) I t is the virtual displacement current between plates d Can use Kirchoffs Rules for NON EQUILIBRIUM situation if one uses displacement current 11
Displacement Current IV Calculation of Induced Magnetic Field due to changing Electric Flux Displacement Current IV + - Id(t) R Ic(t) Ic(t) r E(t) 12
Ampere - Maxwell Law II ( ) ( ) ( ) ( ) ( ) ( ) Ampere -Maxwell Law gives - Ampere - Maxwell Law II ( ) B · d s = m I + I c d choose path in-between plates with radius r there = steady state current I c using Kirchoffs Rule I = I the total displacement in out I ( ) ( ) current at any time t = I t thus d tot c tot p r 2 2 r I ( ) = ( ) ( ) t I t = I t d path p 2 c tot 2 c tot R R 13
Calculation of B field using Ampere - Maxwell Law on this path the magnetic field is constant and Calculation of B field using Ampere - Maxwell Law parallel to the path. Right hand rule for I d B · d s = Bds = B ds = B 2 p r r 2 ( ) = m I + I = m I ( ) t c d R 2 c tot ß r 2 ( ) B 2 p r = m I t R 2 c tot ß m r ( ) B r , t = I ( ) t 2 R 2 c tot 14
Maxwells Equations - Integral form 15
Changing Fields Changing Electric Field Changing Magnetic Field Fluctuating electric and magnetic fields Electro-Magnetic Radiation Changing Magnetic Field 16
Speed of Light 17
Lorentz Force completely describe the behavior Maxwells Equations PLUS the Lorentz Force completely describe the behavior of electricity and magnetism 18
Maxwells Laws - Differential Form I Differential Form of Maxwells equations 19
ò ( ) Derivation I e E · d A = r r dV B · d A = closed surface enclosed volume B · d A = closed surface 20
ò ò Derivation II ¶ B E · d s = - · d A ¶ t é ù ê ú F = B · d A ê ú ê closed path Area enclosed by path é ù ê ú F = B · d A ê ú B ê ë ú û ò æ ö 1 ¶ E B · d s = ç J + e ÷ · d A m è ø ¶ t closed path area enclosed by path é ù ê ú I = J · d A ê ú ë ê ú û 21
Vector Calculus Vector Calculus 22
Gauss' and Stokes Theorems Divergence Theorem ò F · ( d A = Ñ · ) F dV closed surface volume enclosed by surface é æ ö æ ö ¶ ¶ æ ö ù ¶ ê Ñ = ç ÷ i + ç ÷ j + ç ÷ k ú ë è x ø ¶ è ¶ y ø è ¶ z ø û Stokes ' Theorem ò F · ( ) d s = Ñ ´ F · d A closed path area enclosed by path 23
ò ò ò ò Using Theorems I ( ) ( ) ¶ B · = Ñ ´ E d s E · d A = - · d A ¶ closed path area enclosed by path Area enclosed by path ¶ B Þ Ñ ´ E = ¶ t ò ò æ 1 m 1 m ¶ E ö B · d s = ( Ñ ´ ) B · d A = ç J + e ÷ · d A è ¶ t ø closed path area enclosed by path area enclosed by path 1 m ¶ E Þ Ñ ´ B = J + e ¶ t 24
ò ò ò Using Theorems II ( ) ( ) ( ) ( ) e E · d A = e Ñ · E = r d A r dV closed surface enclosed volume enclosed volume r ( ) r Þ Ñ · E = e ò B · d A = ( Ñ · ) B d A = closed surface enclosed volume Þ Ñ · B = 25
Maxwells Equations Maxwells Equations ( ) e Ñ · E = r r Ñ · B = ¶ B Ñ ¶ B Ñ ´ E = ¶ t ¶ E 1 m (Ñ ´ B) = J + e ¶ t 26