1 Electromagnetic TheoryG. Franchetti, GSI CERN Accelerator – School Budapest, 2-14 / 10 / 2016 3/10/16 G. Franchetti

2 Mathematics of EM Fields are 3 dimensional vectors dependent of their spatial position (and depending on time) 3/10/16 G. Franchetti

3 Products Scalar product Vector product 3/10/16 G. Franchetti

4 The gradient operator Is an operator that transform space dependent scalar in vector Example: given 3/10/16 G. Franchetti

5 Divergence / Curl of a vector fieldDivergence of vector field Curl of vector field 3/10/16 G. Franchetti

6 Relations 3/10/16 G. Franchetti

7 Flux Concept Example with water v v v 3/10/16 G. Franchetti

8 v v Volume per second or v θ v L 3/10/16 G. Franchetti

9 Flux θ A 3/10/16 G. Franchetti

10 Flux through a surface 3/10/16 G. Franchetti

11 Flux through a closed surface: Gauss theoremAny volume can be decomposed in small cubes Flux through a closed surface 3/10/16 G. Franchetti

12 Stokes theorem Ex(0,D,0) y Ey(D,0,0) Ey(0,0,0) x Ex(0,0,0) 3/10/16G. Franchetti

13 for an arbitrary surface3/10/16 G. Franchetti

14 How it works 3/10/16 G. Franchetti

15 Electric Charges and ForcesTwo charges + - Experimental facts + + + - - - 3/10/16 G. Franchetti

16 Coulomb law 3/10/16 G. Franchetti

17 Units C2 N-1 m-2 System SI Newton Coulomb Meterspermettivity of free space 3/10/16 G. Franchetti

18 Superposition principle3/10/16 G. Franchetti

19 Electric Field 3/10/16 G. Franchetti

20 force electric field 3/10/16 G. Franchetti

21 By knowing the electric field the force on a charge is completely known3/10/16 G. Franchetti

22 Work done along a path work done by the charge 3/10/16 G. Franchetti

23 Electric potential For conservative field V(P) doesnot depend on the path ! Central forces are conservative UNITS: Joule / Coulomb = Volt 3/10/16 G. Franchetti

24 Work done along a path B A work done by the charge 3/10/16G. Franchetti

25 Electric Field  Electric PotentialIn vectorial notation 3/10/16 G. Franchetti

26 Electric potential by one chargeTake one particle located at the origin, then 3/10/16 G. Franchetti

27 Electric Potential of an arbitrary distributionset of N particles origin 3/10/16 G. Franchetti

28 Electric potential of a continuous distributionSplit the continuous distribution in a grid origin 3/10/16 G. Franchetti

29 Energy of a charge distributionit is the work necessary to bring the charge distribution from infinity More simply More simply 3/10/16 G. Franchetti

30 and the “divergence theorem” it can be proved thatIn integral form Using and the “divergence theorem” it can be proved that is the density of energy of the electric field 3/10/16 G. Franchetti

31 Flux of the electric fieldθ A 3/10/16 G. Franchetti

32 Flux of electric field through a surface3/10/16 G. Franchetti

33 Application to Coulomb lawOn a sphere This result is general and applies to any closed surface (how?) 3/10/16 G. Franchetti

34 On an arbitrary closed curveq2 q1 3/10/16 G. Franchetti

35 First Maxwell Law integral form for a infinitesimal small volumedifferential form (try to derive it. Hint: used Gauss theorem) 3/10/16 G. Franchetti

36 Physical meaning If there is a charge in one place, the electric flux is different than zero One charge create an electric flux. 3/10/16 G. Franchetti

37 Poisson and Laplace EquationsAs and combining both we find Poisson In vacuum: Laplace 3/10/16 G. Franchetti

38 Magnetic Field There exist not a magnetic charge!(Find a magnetic monopole and you get the Nobel Prize) 3/10/16 G. Franchetti

39 Ampere’s experiment I1 I2 3/10/16 G. Franchetti

40 Ampere’s Law dl all experimental results fit with this law I2 B F I3/10/16 G. Franchetti

41 Units From [Tesla] From follows To have 1T at 10 cm with one cableI = 5 x 105 Amperes !! 3/10/16 G. Franchetti

42 Biot-Savart Law dB I From analogy with the electric field 3/10/16G. Franchetti

43 Lorentz force + A charge not in motion does not experience a force ! Bv F No acceleration using magnetic field ! 3/10/16 G. Franchetti

44 3/10/16 G. Franchetti

45 Flux of magnetic field There exist not a magnetic charge ! No matter what you do.. The magnetic flux is always zero! 3/10/16 G. Franchetti

46 Second Maxwell Law Integral form Differential from 3/10/16G. Franchetti

47 Changing the magnetic Flux...Following the path L 3/10/16 G. Franchetti

48 Faraday’s Law integral form valid in any way the magnetic fluxis changed !!! (Really not obvious !!) 3/10/16 G. Franchetti

49 for an arbitrary surface3/10/16 G. Franchetti

50 Faraday’s Law in differential form3/10/16 G. Franchetti

51 Summary Faraday’s Law Integral form differential form 3/10/16G. Franchetti

52 Important consequenceA current creates magnetic field magnetic field create magnetic flux L = inductance [Henry] Changing the magnetic flux creates an induced emf 3/10/16 G. Franchetti

53 r h energy necessary to create the magnetic field 3/10/16G. Franchetti

54 Field inside the solenoidMagnetic flux Therefore Energy density of the magnetic field 3/10/16 G. Franchetti

55 Ampere’s Law I B 3/10/16 G. Franchetti

56 Displacement Current I 3/10/16 G. Franchetti

57 Displacement Current Here there is a varying electric fieldbut no current ! I 3/10/16 G. Franchetti

58 Displacement Current Stationary current I  electricfield changes with time I This displacement current has to be added in the Ampere law 3/10/16 G. Franchetti

59 Final form of the Ampere lawintegral form differential form 3/10/16 G. Franchetti

60 Maxwell Equations in vacuumIntegral form Differential form 3/10/16 G. Franchetti

61 Magnetic potential ? ? Can we find a “potential” such thatMaxwell equation it means that we cannot include currents !! But 3/10/16 G. Franchetti

62 Example: 2D multipoles For 2D static magnetic field in vacuum (only Bx, By) These are the Cauchy-Reimann That makes the function analytic 3/10/16 G. Franchetti

63 Vector Potential In general we require(this choice is always possible) Automatically 3/10/16 G. Franchetti

64 Solution Electric potential Magnetic potential 3/10/16 G. Franchetti

65 Effect of matter Electric field Magnetic field Conductors DiamagnetismParamagnetism Ferrimagnetism Dielectric 3/10/16 G. Franchetti

66 Maxwell equation in vacuum are always valid, even when we consider the effect of matter Microscopic field Averaged field That is the field is “local” between atoms and moving charges this is a field averaged over a volume that contain many atoms or molecules 3/10/16 G. Franchetti

67 Conductors bounded to free electrons be inside the conductor 3/10/16G. Franchetti

68 Conductors and electric fieldbounded to be inside the conductor on the surface the electric field is always perpendicular surface distribution of electrons 3/10/16 G. Franchetti

69 A Applying Gauss theorem 3/10/16 G. Franchetti

70 Boundary condition The surfaces of metals are always equipotential + +- - - - - 3/10/16 G. Franchetti

71 Ohm’s Law [Ω] [Ωm] resistivity free electrons A conductivity l or3/10/16 G. Franchetti

72 Who is who ? 3/10/16 G. Franchetti