* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 7 - web page for staff
Magnetorotational instability wikipedia , lookup
Computational electromagnetics wikipedia , lookup
Electricity wikipedia , lookup
Electromotive force wikipedia , lookup
Friction-plate electromagnetic couplings wikipedia , lookup
Electric machine wikipedia , lookup
Electromagnetism wikipedia , lookup
Neutron magnetic moment wikipedia , lookup
Magnetic field wikipedia , lookup
Magnetic nanoparticles wikipedia , lookup
Maxwell's equations wikipedia , lookup
Hall effect wikipedia , lookup
Superconducting magnet wikipedia , lookup
Magnetometer wikipedia , lookup
Magnetic monopole wikipedia , lookup
Galvanometer wikipedia , lookup
Earth's magnetic field wikipedia , lookup
Lorentz force wikipedia , lookup
Magnetic core wikipedia , lookup
Superconductivity wikipedia , lookup
Multiferroics wikipedia , lookup
Force between magnets wikipedia , lookup
Magnetoreception wikipedia , lookup
Scanning SQUID microscope wikipedia , lookup
Magnetochemistry wikipedia , lookup
Magnetohydrodynamics wikipedia , lookup
Faraday paradox wikipedia , lookup
ENE 325 Electromagnetic Fields and Waves Lecture 7 Ampére’s circuital law 03/05/60 1 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. 03/05/60 2 Review (2) Id L a r Id L R dH 2 4 r 4 r 3 Magnetic field intensity from an infinite length line of current (line is located on z-axis) I a H 2 Magnetic field intensity from a ring of current (a ring is located on x-y plane and the observation point is on z-axis. H 03/05/60 Ia 2 2 h2 a 2 3/ 2 az 3 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. 2 2I H az L Right hand rule 03/05/60 4 Outline Ampére’s circuital law Curl and point form of Ampére’s circuital law Magnetic flux density 03/05/60 5 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. H d L I enc A To find , choose the proper Amperian path that is everywhere either tangential or normal to and over which is constant. 03/05/60 6 Use Ampere’s circuital law to determine H from the infinite line of current. From H d L I enc H H a d L d a 2 then H d L H a d a I enc . 0 H 03/05/60 I 2 a A/m. 7 Magnetic field of the uniform sheet of current (1) K K ax Create path a-b-c-d and perform the integration along the path. 03/05/60 8 Magnetic field of the uniform sheet of current (2) From H d L I enc H y (w) H z (h) H y (w) H z (h) K x w, divide the sheet into small line segments along x-axis, by symmetry Hz is cancelled. . 2H y K x Kx Hy . 2 Because of the symmetry, the magnetic field intensity on one side of the current sheet is the negative of that on the other. 03/05/60 9 Magnetic field of the uniform sheet of current (3) Above the sheet, and or we can write 1 H y1 K x 2 1 H y2 Kx 2 . H (z > 0) (z < 0) 1 A/m K an 2 where a n is a unit vector normal to the current sheet. 03/05/60 10 Magnetic field inside the solenoid a d x h y b z c w From 03/05/60 H d L I enc 11 Magnetic field inside the toroid that has a circular cross section (1) 03/05/60 12 Magnetic field inside the toroid that has a circular cross section (2) From 03/05/60 H d L I enc 13 Ex1 Determine H at point P (0.01, 0, 0) m from two current filaments as shown. 03/05/60 14 Ex2 Determine H 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 03/05/60 15 b) at a < < b c) at b < < c d) at > c 03/05/60 16 Ex3 Determine H at point (10, 0, 0) mm resulted from three current sheets: K1 = 1.5a y ay A/m at x = 6 mm, K2 = -3A/m at x = 9 mm, and K3 = 1.5A/m at x = 12 mm. ay 03/05/60 17 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 H or H is the maximum circulation of H per unit area as the area shrinks to zero. H dL H lim J S 0 S 03/05/60 18 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 H ax H x Hx 03/05/60 ay az . y z Hy Hz 19 Physical view of curl a) Field lines indicating divergence b) Field lines indicating curl 03/05/60 A simple way to see the direction of curl using right hand rule 20 Stokes’s Theorem Stokes’s Theorem relates a closed line integral into a surface integral H d L H d S 03/05/60 21 Magnetic flux density, B Magnetic flux density B is related to the magnetic field intensity H in the free space by B 0 H Weber/m2 or Tesla (T) 1 Tesla = 10,000 Gauss. where 0 is the free space permeability, given in units of henrys per meter, or 0 10-7 H/m. Magnetic flux (units of Webers) passing through a surface is found by B dS 03/05/60 22 Gauss’s law for magnetic fields B dS 0 or 03/05/60 B 0. 23 EX4 A 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 a z 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 03/05/60 24 Maxwell’s equations for static fields 03/05/60 Integral form Differential form D d S Qenc D v B dS 0 B 0 E dL 0 E 0 H d L I enc H J 25