Lesson 12 Maxwells’ Equations Gauss’ Law Faradays’ Law Amperes’ Law Ampere - Maxwell Law Maxwells Equations Integral Form Differential Form 1

Gauss' Law Gauss’ Law For Electric Fields: Q E · A = = F d e surface enclosing electric charge Gauss’ Law For Magnetic Fields: B · = = F d A B surface enclosing magnetic charge 2

Amperes and Faradays Laws Amperes Law Amperes and Faradays Laws ò 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

Faradays Law Faradays Law 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 At each instant of time Maxwells Law of Induction ò ( ) 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 ò 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 8

Displacement Current I 9

Displacement Current II Get varying electric fields in capacitors Ic(t) + - E(t) 10

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 d F 1 dQ d 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

ò Ampere - Maxwell Law II ( ) ( ) ( ) ( ) ( ) ( ) Ampere - Maxwell Law B · s = m + d I I c d choose path inbetween plates with radius r there steady state current = I c = using Kirchoffs Rule I I the total displacement in out ( ) ( ) current at any time I 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 parallel to the path Calculation of B field using Ampere - Maxwell Law right hand rule for I thus d ò ò ò 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 Maxwells Equations PLUS the Lorentz Force completly describe the behaviour of electricity and magnetism 18

Maxwells Laws - Differential Form I Differential Form of Maxwells equations 19

ò ò ò ( ) Derivation I e E · d A = r r dV B · d A = closed surface enclosed volume ò B · d A = closed surface 20

ò ò ò ò ò ò Derivation II ¶ B E · d s = - · d A ¶ t é ù ê ú F = B · d closed path Area enclosed by path é ò ù ê ú F = B · d A ê ú B ê ë ú û ò ò æ ö 1 ¶ E B · d s = ç J + e ÷ · d A m è ¶ ø t closed path area enclosed by path é ò ù ê ú I = J · d A ê ú ê ë û ú 21

Vector Calculus Vector Calculus 22

Gauss' and Stokes Theorems Divergence Theorem ò ò F · d A = ( Ñ · ) F dV closed surface volume enclosed by surface é æ æ ö ¶ ö ¶ æ ù ¶ ö ê Ñ = ç ÷ i + ç ÷ j + ç ÷ k ú ë è ¶ x ø è ¶ y ø è ¶ z ø û Stokes ' Theorem ò ò · ( ) F d s = Ñ ´ F · d A closed path area enclosed by path 23

ò ò ò ò ò ò Using Theorems I ( ) ( ) ¶ B E · d s = Ñ ´ E · d A = - · d closed path area enclosed by path Area enclosed by path ¶ B Þ Ñ ´ E = ¶ t ò ò ò æ ¶ E ö 1 m 1 m ( ) 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 ò ò Using Theorems II ( ) ( ) ( ) ( ) e E · d A = e Ñ · E d A = dV closed surface enclosed volume enclosed volume r ( ) r Þ e Ñ · E = e ò 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