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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: Where; E = electric field intensity D = electric flux density ρv = electric charge density per unit volume H = magnetic field intensity B = magnetic flux density

  4. Maxwell’s equations • For staticcase, ∂/∂t = 0. • Maxwell’s equations is reduced to: ElectrostaticsMagnetostatics

  5. Magnetic Force B = Magnetic Flux Density B B q q I q B B

  6. 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.

  7. 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)

  8. 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:

  9. 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.

  10. Magnetic Force on a Current- Carrying Conductor • On a line segment, Fm is proportional to the vector between the end points.

  11. 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.

  12. Solution to Example 1 • a)

  13. The Biot–Savart’s Law The Biot–Savart law is used to compute the magnetic field generated by a steady current, i.e. a continual flow of charges, for example through a wire Biot–Savart’slawstates that: where: dH = differential magnetic field dl = differential length

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

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

  16. 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.

  17. = dl = -dl O  A C  O A C 0 ? • For segment AC, dl is in φ direction, • Using Biot- Savart’s law:

  18. QUIZ!! z Find E1 if E2 = 2x -3y +3z with s = 3.54 x 10-11(C/m2) And find Ө1 and Ө2 ε1=2ε0 ε2=8ε0 Ө2 E2 xy E1 Ө1

  19. Gauss’s Law for Magnetism • Gauss’s law for magnetismstates that: • Magnetic field lines always form continuous closed loops.

  20. Ampere’s law for magnetism • Ampere’s law states that: • true for an infinite lengthof conductor H C, +aø dl true for an infinite length of conductor I, +az r

  21. Magnetic Field of an infinite length of conductor From then re-arrange the equation in terms of Hφ: Hence, the magnetic field vector, H:

  22. Example 3 • A toroidal coil with Nturns carrying a current I , determine the magnetic field H in each of the following three regions: r < a, a < r < b,andr > 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 • inner conductor radius = r1 0.01m • outer conductor radius = r20.05m • current of 2.5A (in the +azdirection) • Flux radius = 2m Iaz=2.5A z aø Flux,z xy r1 r2

  27. Solution to Example 4 where dS is in the aø direction. 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 The behavior of magnetic dipole moments & magnetic susceptibility, of its atoms with an external magnetic field is used as a basis for classifying magnetic materials.

  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.eorbital magnetic moment, mo • Electron spin about its own axis, i.espin magnetic moment, ms

  31. Magnetic Permeability • Magnetization vectorM is defined as where = magnetic susceptibility (dimensionless) • Magnetic permeability is defined as: and to define the magnetic properties in term of relative permeability is defined as:

  32. Magnetic Materials - Diamagnetic • metals have a very weak and negative susceptibility ( ) to magnetic field • slightly repelled by a magnetic field and the material does not retain the magnetic properties when the external field is removed • Most elements in the periodic table, including copper, silver, and gold, are diamagnetic.

  33. Magnetic Materials - Paramagnetic • Paramagnetic materials have a small and positive susceptibilities to magnetic fields. • slightly attracted by a magnetic field and the material does not retain the magnetic properties when the external field is removed. • Paramagnetic materials include magnesium, molybdenum, lithium, and tantalum.

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

  35. Magnetic Materials – 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.

  36. Process of Magnetic Hysteresis material is magnetized and can serve as permanent magnet! B material is demagnetize

  37. Magnetic Hysteresis of Ferromagnetic Materials • Comparison of hysteresis curves for (a) a hard and (b) a soft ferromagnetic material is shown. Hard magnetic material- cannot be easily magnetized & demagnetized by an external magnetic field. Soft magnetic material – easily magnetized & demagnetized.

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

  39. Magnetic boundary conditions • Boundary between medium 1 with μ1 and medium 2 with μ2

  40. 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: • Since , • For linear, isotropic media, the first boundary condition which is related to H;

  41. z xy By applying Ampere’s law

  42. 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:

  43. Example xy (plane)

  44. Solution: • H1t = H2t thus, H2t = 6ax + 2ay • Hn1 = 3az, • but, Hn2 = ?? • 6000μ0(3az) = 3000 μ0(Hn2) • Hn2 = 6az • thus, H2 =6ax + 2ay + 6az μr1 = 6000 ; μr2 = 3000 ;

  45. 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.

  46. INDUCTANCE store magnetic energy

  47. 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.

  48. 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

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