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

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

3 Amperes and Faradays LawsAmperes 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

4 Faradays Law Faradays LawChange of emf around closed loop due to static Electric Field Change of emf around closed loop due to induced Electric Field 4

5 Changing Magnetic Flux Produces Induced Electric FieldChanging Flux I 5

6 Maxwells Law of InductionAt 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

7 Changing Electric Flux Produces Induced Magnetic FieldChanging Flux II 7

8 We can thus generalize Amperes Law to look exactly analogous to Faradays’ LawAmpere - Maxwell Law I 8

9 Displacement Current I9

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

11 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

12 Displacement Current IVCalculation of Induced Magnetic Field due to changing Electric Flux Displacement Current IV Id(t) R Ic(t) Ic(t) r E(t) 12

13 ò Ampere - Maxwell Law II ( ) ( ) ( ) ( ) ( ) ( ) Ampere - Maxwell LawB 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

14 Calculation of B field using Ampere - Maxwell Lawon 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

15 Maxwells Equations - Integral form15

16 Changing Fields Changing Electric Field Changing Magnetic FieldFluctuating electric and magnetic fields Electro-Magnetic Radiation Changing Magnetic Field 16

17 Speed of Light 17

18 Lorentz Force Maxwells Equations PLUS the Lorentz Forcecompletly describe the behaviour of electricity and magnetism 18

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

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

21 ò ò ò ò ò ò Derivation II ¶ B E · d s = - · d A ¶ t é ù ê ú F = B · dclosed 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

22 Vector Calculus Vector Calculus 22

23 Gauss' and Stokes TheoremsDivergence 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

24 ò ò ò ò ò ò Using Theorems I ( ) ( ) ¶ B E · d s = Ñ ´ E · d A = - · dclosed 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

25 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

26 Maxwells Equations Maxwells Equations ( ) e Ñ · E = r r Ñ · B = ¶ B ÑB Ñ E = t E 1 m (Ñ B) = J + e t 26