ENE 325 Electromagnetic Fields and Waves

1. ENE 325Electromagnetic Fields and Waves Lecture 7 Ampére’s circuital law

2. Review (1) • Capacitance • Static magnetic field (time invariant) • Source of magnetic field • permanent magnet • electric field changing linearly with time • direct current • Bio-Savart law is a method to determine the magnetic field intensity. It is an analogy to Coulomb’s law of Electrostatics.

3. Review (2) • Magnetic field intensity from an infinite length line of current(line is located on z-axis) • Magnetic field intensity from a ring of current (a ring is located on x-y plane and the observation point is on z-axis.

4. Review (3) • Magnetic field intensity from a rectangular loop of current (a wire loop is located on x-y plane and the observation point is at the origin. • Right hand rule

5. Outline • Ampére’s circuital law • Curl and point form of Ampére’s circuital law • Magnetic flux density

6. Ampére’s circuital law • Analogy to Gauss’s law • Use for magnetostatic’s problems with sufficient symmetry. • Ampere’s circuital law – the integration of around any closed path is equal to the net current enclosed by that path. • To find , choose the proper Amperian path that is everywhere either tangential or normal to and over which is constant. A

7. Use Ampere’s circuital law to determine from the infinite line of current. From then A/m.

8. Magnetic field of the uniform sheet of current (1) Create path a-b-c-d and perform the integration along the path.

9. Magnetic field of the uniform sheet of current (2) From divide the sheet into small line segments along x-axis, by symmetry Hzis cancelled. . Because of the symmetry, the magnetic field intensity on one side of the current sheet is the negative of that on the other.

10. Magnetic field of the uniform sheet of current (3) Above the sheet, (z > 0) (z < 0) and . or we can writeA/m where is a unit vector normal to the current sheet.

11. Magnetic field inside the solenoid From

12. Magnetic field inside the toroid that has a circular cross section (1) consisting of a circular ring-shaped magnetic core of iron powder, ferrite, or other material around which wire is coiled to make an inductor. The magnetic flux in a toroid is largely confined to the core, preventing its energy from being absorbed by nearby objects, making toroidal cores essentially self-shielding.

14. Magnetic field inside the toroid that has a circular cross section (2) From

15. Ex1 Determine at point P (0.01, 0, 0) m from two current filaments as shown.

16. Ex2 Determine for the coaxial cable that has a inner radius a = 3 mm, b = 9 mm, and c = 12 mm. Given I = 0.8 A. a) at  < a

17. b) at a <  < b c) at b <  < c d) at  > c

18. Ex3 Determine at point (10, 0, 0) mm resulted from three current sheets: K1 = 1.5 A/m at x = 6 mm, K2 = -3 A/m at x = 9 mm, and K3 = 1.5 A/m at x = 12 mm.

19. Curl and the point form of Ampére’s circuital law (1) • ‘Curl’ is employed to find the point form Ampère’s circuital law, analogous to ‘Divergence’ to find the point form of Gauss’s law. • Curl of or is the maximum circulation of per unit area as the area shrinks to zero. • It gives a measure of the degree to which a field curls around a particular point.

20. Curl and the point form of Ampére’s circuital law (2) • ‘Curl´ operator perform a derivative of vector and returns a vector quantity. For Cartesian coordinates, can be written as

21. Physical view of curl • Field lines indicating divergence A simple way to see the • Field lines indicating curl direction of curl using • right hand rule

22. Stokes’s Theorem • Stokes’s Theorem relates a closed line integral into a surface integral

23. Magnetic flux density, B • Magnetic flux density is related to the magnetic field intensity in the free space by • Magnetic flux (units of Webers) passing through a surface is found by Weber/m2 or Tesla (T) 1 Tesla = 10,000 Gauss. The earth’s B is about 0.5 G. where 0 is the free space permeability, given in units of henrys per meter, or 0 = 410-7 H/m.

24. A fundamental feature of magnetic fields that distinguishes them from electric fields is that the field lines form closed loops. Gaussian Surface

25. You cannot saw magnet in half to isolate the north and the south poles; if you saw magnet in half you get two magnets. Hence you cannot isolate a magnetic pole.

26. Gauss’s law for magnetic fields or The net magnetic flux passing through a Gaussian surface (a closed surface) must be zero. This is also referred to as the law of conservation of magnetic flux.

27. EX4A solid conductor of circular cross section is made of a homogeneous nonmagnetic material. If the radius a = 1 mm, the conductor axis lies on the z axis, and the total current in the direction is 20 A, find a) H at  = 0.5 mm b) B at  = 0.8 mm c) The total magnetic flux per unit length inside the conductor

28. Maxwell’s equations for static fields Integral formDifferential form

30. A parallel plate capacitor with a 1.0 m2 surface area for each plate, a 2.0 mm plate separation, and a dielectric with relative permittivity of 1200 has a 12. V potential difference across the plates. Calculate (a) the capacitance, and (b) the magnitude of the charge density on one of the plates.