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EKT 241/4: ELECTROMAGNETIC THEORY

EKT 241/4: ELECTROMAGNETIC THEORY. UNIVERSITI MALAYSIA PERLIS. CHAPTER 4 – MAGNETOSTATICS. PREPARED BY: NORDIANA MOHAMAD SAAID dianams@unimap.edu.my. Chapter Outline. Maxwell’s Equations Magnetic Forces and Torques The total electromagnetic force, known as Lorentz force Biot- Savart’s law

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EKT 241/4: ELECTROMAGNETIC THEORY

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

  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/m2 • Where: μ = magnetic permeability • Where; • J = current density • H = magnetic field intensity • B = magnetic flux density

  4. Magnetic Forces and Torques • The electric force Fe 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 forceFm 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 Fm is: • In a uniform magnetic field, Fm is zero for a closed circuit.

  7. Magnetic Force on a Current- Carrying Conductor • On a line segment, Fm 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 F1 on the straight section of the wire and (b) the force F2 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 lawstates 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 magnetismstates 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. I Magnetic Field of an infinite length of conductor • Consider a conductor lying on the z axis, carrying current I in +az direction. • Using Ampere’s law: • The path to evaluate is along the aφ direction, hence use dLφ.

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

  21. Magnetic Field of an infinite length of conductor Integrating and then re-arrange the equation in terms of Hφ: Hence, the magnetic field vector, H: Note: this equation is true for 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 < a, a < r < b,and 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/(2πr) .

  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 +azdirection. 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 aφ direction.

  27. Solution to Example 4 So, Therefore,

  28. Vector Magnetic Potential • For any vector of vector magnetic potentialA: • We are able to derive: . • Vector Poisson’s equationis 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, mo • Electron spin about its own axis, i.e spin magnetic moment, ms

  31. Magnetic Permeability • Magnetization vectorM 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 μ1 and medium 2 with μ2

  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 ρs=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 Js=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, Js=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, θ1≈−90° and θ2≈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, Λis 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 B1 generated by current I1 results in a flux Φ12 through loop 2: • If loop 2 consists of N2 turns all coupled by B1 in exactly the same way, the total magnetic flux linkage through loop 2 due to B1 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

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