Electromagnetic Theory G Franchetti GSI CERN Accelerator School

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

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

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

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

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

Relations 3/10/16 G. Franchetti 6

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

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

Flux θ A 3/10/16 G. Franchetti 9

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

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

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

for an arbitrary surface 3/10/16 G. Franchetti 13

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

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

Coulomb law 3/10/16 G. Franchetti 16

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

Superposition principle 3/10/16 G. Franchetti 18

Electric Field 3/10/16 G. Franchetti 19

force electric fie ld 3/10/16 G. Franchetti 20

By knowing the electric field the force on a charge is completely known 3/10/16 G. Franchetti 21

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

Electric potential For conservative field V(P) does not depend on the path ! Central forces are conservative UNITS: Joule / Coulomb = Volt 3/10/16 G. Franchetti 23

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

Electric Field Electric Potential In vectorial notation 3/10/16 G. Franchetti 25

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

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

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

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

In 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 30

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

Flux of electric field through a surface 3/10/16 G. Franchetti 32

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

On an arbitrary closed curve q 2 q 1 3/10/16 G. Franchetti 34

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

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 36

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

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

Ampere’s experiment I 1 I 2 3/10/16 G. Franchetti 39

Ampere’s Law dl all experimental results fit with this law I 2 B F I 3/10/16 G. Franchetti 40

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

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

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

3/10/16 G. Franchetti 44

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 45

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

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

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

for an arbitrary surface 3/10/16 G. Franchetti 49

Faraday’s Law in differential form 3/10/16 G. Franchetti 50

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 66

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

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

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

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

Ohm’s Law [Ω] resistivity free electrons A conductivity or l 3/10/16 [Ωm] G. Franchetti 71

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