1. Lesson 12 Maxwells’ Equations
Gauss’ 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 e
surface enclosing electric charge
Gauss’ Law
For Magnetic Fields: B = = F d A B
surface enclosing magnetic charge 2

3. 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

4. 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

5. Changing Magnetic Flux Produces Induced Electric Field
Changing Flux I 5

6. 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

7. Changing Electric Flux Produces Induced Magnetic Field
Changing Flux II 7

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

9. Displacement Current I9

10. Displacement Current II
Get 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 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

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 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

15. Maxwells Equations - Integral form15

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

17. Speed of Light 17

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

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

20. 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

21. 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

22. Vector Calculus
Vector Calculus 22

23. 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

24. 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

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