5. Magnetostatics

1. 5. Magnetostatics Applied EM by Ulaby, Michielssen and Ravaioli

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. Chapter 5 Overview

4. Course Outcome 3 (CO3) • Ability to analyze the concept of electric current density and boundary conditions, magnetic flux and magnetic flux density in a steadymagnetic field and the basic laws of magnetic fields.

5. 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

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

7. Electric vs Magnetic Comparison

8. Electric & Magnetic Forces Magnetic force Electromagnetic (Lorentz) force

9. Magnetic Force on a Current Element Differential force dFm on a differential current I dl:

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

11. 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)

12. 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:

13. 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.

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

15. 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.

16. Solution to Example 1 • a) the magnetic force F1 on the straight section of the wire

18. Torque d = moment arm F = force T = torque

19. Magnetic Torque on Current Loop No forces on arms 2 and 4 ( because I and B are parallel, or anti-parallel) Magnetic torque: Area of Loop

20. Inclined Loop For a loop with N turns and whose surface normal is at angle theta relative to B direction:

21. 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’s lawstates that: where: dH = differential magnetic field dl = differential length

22. Biot-Savart Law Magnetic field induced by a differential current: For the entire length:

23. Magnetic Field due to Current Densities

24. 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.

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

26. Example 5-2: Magnetic Field of Linear Conductor Cont.

27. Example 5-2: Magnetic Field of Linear Conductor

28. Magnetic Field of Long Conductor

30. Example 5-3: Magnetic Field of a Loop Magnitude of field due to dl is dH is in the r–z plane , and therefore it has components dHr and dHz z-components of the magnetic fields due to dl and dl’ add because they are in the same direction, but their r-components cancel Hence for element dl: Cont.

31. Example 5-3:Magnetic Field of a Loop (cont.) For the entire loop:

32. Magnetic Dipole Because a circular loop exhibits a magnetic field pattern similar to the electric field of an electric dipole, it is called a magnetic dipole

33. Forces on Parallel Conductors Parallel wires attract if their currents are in the same direction, and repel if currents are in opposite directions

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

36. Ampère’s Law

37. 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

38. Internal Magnetic Field of Long Conductor For r < a Cont.

39. External Magnetic Field of Long Conductor For r > a

40. Magnetic Field of Toroid Applying Ampere’s law over contour C: Ampere’s law states that the line integral of H around a closed contour C is equal to the current traversing the surface bounded by the contour. The magnetic field outside the toroid is zero. Why?

41. 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:

42. 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.

43. 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

44. Solution to Example 4 where dS is in the aø direction. So, Therefore,

45. Magnetic Vector Potential A Electrostatics Magnetostatics

46. Vector Magnetic Potential • For any vector of vector magnetic potentialA: • We are able to derive: . • Vector Poisson’s equationis given as: where

47. 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

48. Magnetic Properties of Materials

49. 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:

50. 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.