1. ELECTROMAGNETIC THEORY EKT 241/4: ELECTROMAGNETIC THEORY PREPARED BY: NORDIANA MOHAMAD SAAID dianams@unimap.edu.my CHAPTER 4 – MAGNETOSTATICS

2. Chapter Outline Maxwell’s Equations Magnetic Forces and Torques The total electromagnetic force, known as Lorentz force Biot- Savart’s law Gauss’s law for magnetism Ampere’s law for magnetism Magnetic Field and Flux Vector magnetic potential Properties of 3 different types of material Boundary conditions between two different media Self inductance and mutual inductance Magnetic energy

3. Maxwell’s equations Maxwell’s equations for magnetostatics: Relationship between B and H: unit: Tesla or Weber/m 2 Where: μ = magnetic permeability Where; J = current density H = magnetic field intensity B = magnetic flux density

4. Magnetic Forces and Torques The electric force F e per unit charge acting on a test charge placed at a point in space with electric field E. When a charged particle moving with a velocity u passing through that point in space, the magnetic force F m is exerted on that charged particle. where B = magnetic flux density (Cm/s or Tesla T)

5. Magnetic Forces and Torques If a charged particle is in the presence of both an electric field E and magnetic field B, the total electromagnetic force acting on it is:

6. Magnetic Force on a Current- Carrying Conductor For closed circuit of contour C carrying I, total magnetic force F m is: In a uniform magnetic field, F m is zero for a closed circuit.

7. Magnetic Force on a Current- Carrying Conductor On a line segment, F m is proportional to the vector between the end points.

8. Example 1 The semicircular conductor shown carries a current I. The closed circuit is exposed to a uniform magnetic field. Determine (a) the magnetic force F 1 on the straight section of the wire and (b) the force F 2 on the curved section.

9. Solution to Example 1 a) b)

10. Magnetic Torque on a Current- Carrying Loop Applied force vector F and distance vector d are used to generate a torque T T = d× F (N·m) Rotation direction is governed by right-hand rule.

11. The Biot–Savart’s Law Biot–Savart’s law states that: where: dH = differential magnetic field dI = differential current element

12. The Biot–Savart’s Law To determine the total H:

13. The Biot–Savart’s Law Biot–Savart’s law may be expressed in terms of distributed current sources.

14. Example 2 Determine the magnetic field at the apex O of the pie-shaped loop as shown. Ignore the contributions to the field due to the current in the small arcs near O.

15. Solution o Example 2 For segment OA and OC, the magnetic field at O is zero since is parallel and anti-parallel to. For segment AC, dl is in φ direction, Using Biot- Savart’s law:

16. Magnetic Force between Two Parallel Conductors Force per unit length on parallel current-carrying conductors is: where F’ 1 = -F’ 2 (attract each other with equal force)

17. Gauss’s Law for Magnetism Gauss’s law for magnetism states that: Magnetic field lines always form continuous closed loops.

18. Ampere’s law for magnetism Ampere’s law states that: The directional path of current C follows the right-hand rule.

19. Magnetic Field of an infinite length of conductor Consider a conductor lying on the z axis, carrying current I in +a z direction. Using Ampere’s law: The path to evaluate is along the a φ direction, hence use dL φ. I

20. Using Ampere’s law: Where; Thus, Magnetic Field of an infinite length of conductor

21. Integrating and then re-arrange the equation in terms of H φ : Hence, the magnetic field vector, H: Note: this equation is true for an infinite length of conductor Magnetic Field of an infinite length of conductor

22. Example 3 A toroidal coil with N turns carrying a current I, determine the magnetic field H in each of the following three regions: r b, all in the azimuthal plane of the toroid.

23. Solution to Example 3 H = 0 for r < a as no current is flowing through the surface of the contour H = 0 for r > b, as equal number of current coils cross the surface in both directions. For a < r < b, we apply Ampere’s law: Hence, H = NI/(2 πr ).

24. Magnetic Flux The amount of magnetic flux, φ in Webers from magnetic field passing through a surface is found in a manner analogous to finding electric flux:

25. Example 4 An infinite length coaxial cable with inner conductor radius of 0.01m and outer conductor radius of 0.05m carrying a current of 2.5A exists along the z axis in the +a z direction. Find the flux passing through the region between two conductors with height of 2 m in free space.

26. Solution to Example 4 The relation between B and H is: To find magnetic flux crossing the region, we use: unit: Weber where dS is in the a φ direction.

27. Solution to Example 4 So, Therefore,

28. Vector Magnetic Potential For any vector of vector magnetic potential A: We are able to derive:. Vector Poisson’s equation is given as: where

29. Magnetic Properties of Materials Magnetic behavior of a material is due to the interaction of magnetic dipole moments of its atoms with an external magnetic field. This behavior is used as a basis for classifying magnetic materials. 3 types of magnetic materials: diamagnetic, paramagnetic, and ferromagnetic.

30. Magnetic Properties of Materials Magnetization in a material is associated with atomic current loops generated by two principal mechanisms: –Orbital motions of the electrons around the nucleus, i.e orbital magnetic moment, m o –Electron spin about its own axis, i.e spin magnetic moment, m s

31. Magnetic Permeability Magnetization vector M is defined as where = magnetic susceptibility (dimensionless) Magnetic permeability is defined as: and relative permeability is defined as

32. Magnetic Materials Diamagnetic materials have negative susceptibilities. Paramagnetic materials have positive susceptibilities. However, the absolute susceptibilities value of both materials is in the order 10 -5. Thus, can be ignored. Hence, we have Diamagnetic and paramagnetic materials include dielectric materials and most metals.

33. Magnetic Hysteresis of Ferromagnetic Materials Ferromagnetic materials is characterized by magnetized domain - a microscopic region within which the magnetic moments of all its atoms are aligned parallel to each other. Hysteresis – “to lag behind”. It determines how easy/hard for a magnetic material to be magnetized and demagnetized. Hard magnetic material- cannot be easily demagnetized by an external magnetic field. Soft magnetic material – easily magnetized & demagnetized.

34. Magnetic Hysteresis of Ferromagnetic Materials Properties of magnetic materials as follows:

35. Magnetic Hysteresis of Ferromagnetic Materials Comparison of hysteresis curves for (a) a hard and (b) a soft ferromagnetic material is shown.

36. Magnetic boundary conditions Boundary between medium 1 with μ 1 and medium 2 with μ 2

37. Magnetic boundary conditions Boundary condition related to normal components of the electric field; By analogy, application of Gauss’s law for magnetism, we get first boundary condition: i.e the normal component of B is continuous across the boundary between two adjacent media

38. Magnetic boundary conditions Since, For linear, isotropic media, the first boundary condition which is related to H; Reversal concept: whereas the normal component of B is continuous across the boundary, the normal component of D (electric flux density) may not be continuous (unless ρ s =0)

39. Magnetic boundary conditions A similar reversal concept applies to tangential components of the electric field E and magnetic field H. Reversal concept related to tangential components: –Whereas the tangential component of E is continuous across the boundary, the tangential component of H may not be continuous (unless J s =0). By applying Ampere’s law and using the same method of derivation as for electric field E:

40. Magnetic boundary conditions The result is generalized to a vector form: Where However, surface currents can exist only on the surfaces of perfect conductors and perfect superconductors (infinite conductivities). Hence, at the interface between media with finite conductivities, J s =0. Thus:

41. Inductance An inductor is the magnetic analogue of an electrical capacitor. Capacitor can store electric energy in the electric field present in the medium between its conducting surfaces. Inductor can store magnetic energy in the volume comprising the inductors.

42. Inductance Example of an inductor is a solenoid - a coil consisting of multiple turns of wire wound in a helical geometry around a cylindrical core.

43. Magnetic Field in a Solenoid For one cross section of solenoid, When l >a, θ 1 ≈−90° and θ 2 ≈90°, Where, N=nl =total number of turns over the length l

44. Self Inductance Magnetic flux, linking a surface S is given by: In a solenoid with uniform magnetic field, the flux linking a single loop is:

45. Self Inductance Magnetic flux linkage, Λ is the total magnetic flux linking a given conducting structure. Self-inductance of any conducting structure is the ratio of the magnetic flux linkage, Λ to the current I flowing through the structure.

46. Self Inductance For a solenoid: For two conductor configuration:

47. Mutual Inductance Mutual inductance – produced by magnetic coupling between two different conducting structures.

48. Mutual Inductance Magnetic field B 1 generated by current I 1 results in a flux Φ 12 through loop 2 : If loop 2 consists of N 2 turns all coupled by B1 in exactly the same way, the total magnetic flux linkage through loop 2 due to B 1 is:

49. Mutual Inductance Hence, the mutual inductance:

50. Magnetic Energy Consider an inductor with an inductance L connected to a current source. The current I flowing through the inductor is increased from zero to a final value I. The energy expended in building up the current in the inductor: i.e the magnetic energy stored in the inductor

51. Magnetic Energy Magnetic energy density (for solenoid): i.e magnetic energy per unit volume