1. Electromagnetic Theory
G. Franchetti, GSI
CERN Accelerator – School
Budapest, 2-14 / 10 / 2016
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2. Mathematics of EM
Fields are 3 dimensional vectors dependent of their spatial position
(and depending on time)
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3. Products
Scalar product
Vector product
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4. The gradient operator
Is an operator that transform space dependent scalar in vector
Example: given
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5. Divergence / Curl of a vector field
Divergence of vector field
Curl of vector field
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6. Relations
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7. Flux Concept
Example with water v v v
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8. v v
Volume per second or v θ v L
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9. Flux θ A
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10. Flux through a surface
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11. Flux through a closed surface: Gauss theorem
Any volume can be decomposed in
small cubes
Flux through a
closed surface
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12. Stokes theorem Ex(0,D,0) y Ey(D,0,0) Ey(0,0,0) x Ex(0,0,0) 3/10/16
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13. for an arbitrary surface
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14. How it works
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15. Electric Charges and Forces
Two charges + -
Experimental facts + + + - - -
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16. Coulomb law
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17. Units C2 N-1 m-2 System SI Newton Coulomb Meters
permettivity of free space
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18. Superposition principle
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19. Electric Field
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20. force
electric field
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21. By knowing the electric field the force on a charge is completely known
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22. Work done along a path
work done by the charge
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23. Electric potential For conservative field V(P) does
not depend on the path !
Central forces are conservative
UNITS: Joule / Coulomb = Volt
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24. Work done along a path B A work done by the charge 3/10/16
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25. Electric Field  Electric Potential
In vectorial notation
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26. Electric potential by one charge
Take one particle located at the origin, then
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27. Electric Potential of an arbitrary distribution
set of N particles
origin
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28. Electric potential of a continuous distribution
Split the continuous distribution in a grid
origin
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29. Energy of a charge distribution
it is the work necessary to bring the charge distribution from infinity
More simply
More simply
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30. and the “divergence theorem” it can be proved that
In integral form
Using
and the “divergence theorem” it can be proved that
is the density of
energy of the electric
field
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31. Flux of the electric fieldθ A
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32. Flux of electric field through a surface
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33. Application to Coulomb law
On a sphere
This result is general and
applies to any closed surface
(how?)
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34. On an arbitrary closed curveq2 q1
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35. First Maxwell Law integral form for a infinitesimal small volume
differential form
(try to derive it. Hint: used Gauss theorem)
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36. Physical meaning
If there is a charge in one place, the electric flux is different than zero
One charge create an electric flux.
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37. Poisson and Laplace EquationsAs
and
combining both we find
Poisson
In vacuum:
Laplace
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38. Magnetic Field There exist not a magnetic charge!
(Find a magnetic monopole and you get the Nobel Prize)
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39. Ampere’s experiment I1 I2
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40. Ampere’s Law dl all experimental results fit with this law I2 B F I
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41. Units From [Tesla] From follows To have 1T at 10 cm with one cable
I = 5 x 105 Amperes !!
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42. Biot-Savart Law dB I From analogy with the electric field 3/10/16
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43. Lorentz force + A charge not in motion does not experience a force ! Bv F
No acceleration using magnetic field !
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44. 3/10/16
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45. Flux of magnetic field
There exist not a magnetic charge ! No matter what you do..
The magnetic flux is always zero!
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46. Second Maxwell Law Integral form Differential from 3/10/16
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47. Changing the magnetic Flux...
Following the path L
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48. Faraday’s Law integral form valid in any way the magnetic flux
is changed !!!
(Really not obvious !!)
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49. for an arbitrary surface
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50. Faraday’s Law in differential form
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51. Summary Faraday’s Law Integral form differential form 3/10/16
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52. Important consequence
A current creates magnetic field
magnetic field create magnetic flux
L = inductance [Henry]
Changing the magnetic flux creates
an induced emf
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53. r h energy necessary to create the magnetic field 3/10/16
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54. Field inside the solenoid
Magnetic flux
Therefore
Energy density
of the magnetic
field
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55. Ampere’s Law I B
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56. Displacement Current I
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57. Displacement Current Here there is a varying electric field
but no current ! I
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58. Displacement Current Stationary current I  electric
field changes with time I
This displacement current
has to be added in the Ampere
law
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59. Final form of the Ampere law
integral form
differential form
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60. Maxwell Equations in vacuum
Integral form
Differential form
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61. Magnetic potential ? ? Can we find a “potential” such that
Maxwell equation
it means that we cannot
include currents !!
But
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62. Example: 2D multipoles
For 2D static magnetic field in vacuum (only Bx, By)
These are the Cauchy-Reimann
That makes the function
analytic
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63. Vector Potential In general we require
(this choice is always possible)
Automatically
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64. Solution
Electric potential
Magnetic potential
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65. Effect of matter Electric field Magnetic field Conductors Diamagnetism
Paramagnetism
Ferrimagnetism
Dielectric
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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
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67. Conductors bounded to free electrons be inside the conductor 3/10/16
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68. Conductors and electric field
bounded to
be inside the
conductor
on the surface
the electric field is
always perpendicular
surface distribution
of electrons
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69. A
Applying Gauss theorem
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70. Boundary condition The surfaces of metals are always equipotential + +- - - - -
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71. Ohm’s Law [Ω] [Ωm] resistivity free electrons A conductivity l or
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72. Who is who ?
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