Stationary electric field in conducting medium

1. Stationary electric field in conducting medium

2. γ - the conductivity of the conductor This field is described by two vectorsi l - electric field intensity E J - current density ΔSn The stationary electric field in conducting medium will be considered. The constant current is flowing in this conducting material.

3. ΔS ΔV J Δi Δ l Δi Δq

4. Equations of electric field in conducting medium Differential form integral form

5. S i5 i1 i4 J4 i2 i3 dS4 Generalized Kirchhoff’s laws KCL – the current flowing through a closed surface =0

6. C Et dl E KVL – the sum of the voltages along any closed curve =0

7. Ohm’s law formulated in field terms Vector form of Ohm’s law Let’s derive Ohm’s law, which expresses the proportionality of the current flowing through the resistance and the voltage across them. R

8. φ+Δφ φ ΔS ΔVol. ΔS Δl Joule’s law Electric energy transforms into heat in the conductor. p – the volumepower density

9. The continuity of vectors E and J components E1 E1t=E2t J1n=J2n E2 Joule’s law in „field form”

10. Analogy: Electric field Electric circuits OHM’s law KCL KVL JOULE`s law

11. -i i B A MET MET Stationary electric field in conducting medium Resistance of transition

12. i R1 R2 Problem: Let’s calculate the resistance of transition Rt of the cable isolation. The cable is placed in the ground.

13. Let’s assume the cable length l=1m or l=1km, then we receive the resistance per unit of cable length The capacity between the working wire and the screening coat or ground is the same as for the cylindrical capacitor. The parameter characterising a cable is to be found in cable catalogues:

14. Resistance of an isolated electrode: When we move the electrode B to infinity:

15. I R vAB r Earth electrodes A B

17. I R r • Current density B. Electric field intensity C.Potential with respect of the reference point

18. φ(r) φ(R) vAB r rA rB R

19. We can calculate the radius of dangerous zone around the electrode from the inequality: Potential of earth electrode:

20. Earth electrode buried very deep. i

21. i1 i h h h h i i2 Spherical earth electrode dug in under the ground surface on a finite depth h. +

22. i h h i Let’s assume that upper medium is nonconducting then There is only one model because in nonconducting medium this field doesn’t exist.

23. i i i h h i i A deep pit has appeared near the electrode a a h h h a a a

24. I Magnetostatic field Magnetostatic field exists around permanent magnets and wires with the constant current. Equations of magnetostatic field:

25. Fundamentals of Magnetostatic Fields • The effects of magnetic fields known for alomost three milllennia • certain stones attract iron • a large deposit of these stones (“lodestones”) found in the district of Magnesia in Asia Minor (Fe3O4) • first scientific study written in 1600 by William Gilbert • early nineteenth century  Hans Christian Oersted  an electric current in a wire affected a magnetic compass needle • Oersted, Ampere, Gauss, Henry, Faraday, and others raised the magnetic field to equal partner status with the electric field (confirmed later by Maxwell)

26. Basic properties: (1) • Cutting of a large permanent magnet creates a number of smaller permanent magnets • A magnetic monopole has not yet been observed to exists in nature  impossible to separate the north pole from the south pole of magnet

27. This field is described with the pair of vectors: B and H Definition of magnetic induction results from the Lorenz force:

28. Definition of B The vector B numerical value is given by: For defining the vector B we assume that only magnetic field exists, E=0. Because B is the vector it has a length and a direction, hence its definition has parts. Vector Bdirection is determined by such vector of charge velocity, where there is no force acting on the charge. B is the magnetic flux density (inductance), the SI unit:  tesla (T).

29. Moving electron circulating about positive nucleus can be modeled as small electric dipole • It can be also interpreted as being current  hence (Oersted theory) the atom can be thought of as being small magnetic dipole • These dipoles will be assumed to be oriented randomly : Definition of magnetic field intensity (or magnetic field) is theresult of magnetization phenomenon.

30. What will happen to these magnetic fields of individual atoms if external magnetic field is applied to the material? It depends on the type of considered material. Def. is the vector of magnetization magnetic flexibility or susceptibility

32. Three classes of materials: • Diamagnetic materials • Bismuth, copper, diamond,, gold, silver, silicon, lead, mercury • Paramagnetic materials • Aluminium, oxygen, magnesium, titanium, … • Ferromagnetic materials • iron, nickel, cobalt

33. Diamagnetic materials • The magnetic dipoles get reoriented such that their magnetic dipole moments m are in slight opposition to the applied magnetic field B • Without external field magnetic moments of rotating about the positive nucleus cancels magnetic field creating by the spin of the electron • External field perturbs the velocities of the orbiting electrons  small magnetic moment for the atom is created (Lenz’s law) opposite to applied field.

34. Paramagnetic materials • Without external field magnetic moments of rotating about the positive nucleus do not cancel magnetic field creating by the spin of the electron completely leaving the atom with small net magnetic moment. • External field tends to align these moments in the direction of the applied field.

35. Ferromagnetic materials • Domains containing 1015 atoms, each has all of the magnetic domains oriented in the same direction (without external field) • Net magnetization = 0. • With external field domains having their magnetic fields aligned with the external one grow at the expense of the other. • If the external field is small this process will reverse itself • Strong enough field  the domains rotate in the external field direction and collective direction of the domains will remain fixed.

36. Differential form Integral form

37. S c i1 i2 i3 Ampère’s circuital law (or flow law or Ampère's law) This current is called a flow c1 The turn of line c (right side of equation) and the signs of currents on the left side must be in agreement. When we turn the right-hand screw into the line c than we will write currents flowing according to the turn motion of the screw with „+”.

38. André-Marie Ampère 1775 - 1836 Authorof the circuital laws

39. We will prove that vector has only one component i R r Magnetic fieldcaused by very long straight wire with the current i Cylindrical system of coordinates will be used because this problem has a cylindrical symmetry.

40. Field doesn’t depend on the angular coordinate. It is obvious result of cylindrical symmetry. hence So Maxwell’s equation has the form:

41. i R r Only one component of the field exists. Hθ depends on one variable r. • Conclusion: • The current flowing in a very long streight wire generates around this wire magnetic field which has only one component • perpendicular to the direction of the current density vector J, • tangential to the circles concentric with the wire.

42. i R r H Problem: determine field H distribution for

43. H(r) Hmax r R

44. (*) is called magnetic flux. Def. The notion: Equation(*)expresses Kirchhoff’s law for magnetic flux: the sum of magnetic fluxespassing through the closed surface equals zero.

45. S

46. ΔS ΔV J Δi Δ l Electrodynamic force

47. Electrodynamic force is acting on that element of the wire which is situated in magnetic field (determined by the induction B)

48. l dl B Magnitude of this force is: i

49. i1 i2 H21 H12 Let’s consider two very long streight parallel wires with the currents i1 and i2. The distance between wires is a. F12 F21

50. H12 i1 i2 H21 F12 F21