ELEC ELECTROMAGNETIC APPLICATIONS PART B. STATIC ELECTRIC AND MAGNETIC FIELDS (Low frequency) F. Rahman Room EE133

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1 ELEC ELECTROMAGNETIC APPLICATIONS PART B STATIC ELECTRIC AND MAGNETIC FIELDS (Low frequency) F. Rahman Room EE133 Tel: Lecture 1 Introduction & recap on 1 F. Rahman

2 Lecture 1 APPLICATIONS POWER APPARATUS: Capacitors, Inductors, Resistors, Insulators, Cables, Transformers, Power supplies, Motors, Generators, Solenoids, Mechanisms for memory access, Material handling devices, Transportation, Robotics and Automation. OTHER: Semiconductor devices (e.g., transistors, and diodes), various industrial processes requiring electric and magnetic fields, e.g., MRI, CT scanning, Separation and extraction processes and so on. Part B of this course will touch upon items in italics. Lecture 1 Introduction & recap on 2 F. Rahman

3 Low Frequency Applications Wavelength λ >> circuit dimensions 50 Hz : λ = 1,500 km Power & medical 1 MHz : λ = 300 m applications 10 GHz : λ = 30 mm Telecom applications Lecture 1 Introduction & recap on 3 F. Rahman

4 ELECTROMAGNETIC FIELD MODEL Electric Field Intensity, E - V/m Electric Flux Density, D - C/m 2 Magnetic Field or Flux Density, B - T (Tesla) Magnetic Field Intensity, H - A/m Q, J, V E,D,B,H analysis design Lecture 1 Introduction & recap on 4 F. Rahman

5 Source Quantities Volume Charge: ρv = Lim v 0 Surface Charge: ρs = Lim s 0 q v q s C/m 2 C/m 3 Line Charge: ρl = Lim l 0 q l C/m q = Lim = t Current: t 0 I dq dt A Volume current density: J = Lim s 0 I S A/m 2 Surface current density: J I = Lim A/m L s L 0 Voltage: V ab a V = E dl b Lecture 1 Introduction & recap on 5 F. Rahman

6 Chapter 2 Review of some Read this chapter for a good understanding of how is used to describe vector (electric and magnetic) fields in a three dimensional space. Line, surface and volume integrals Cartesian, cylindrical and spherical coordinate systems The (del) operator a + a + a x y z = x y z Gradient of a Scalar Field V = grad V = V V V a x + a y + a z x y z This gradient is a vector point function. Lecture 1 Introduction & recap on 6 F. Rahman

7 Divergence of a Vector Field Div A = i A = flux of A through a closed surface/unit volume Lim = v 0 Aids v A non zero divergence would indicate the presence of a source or a sink. Its value is a measure of the strength of the source or the sink. Divergence of a field is conveniently used to describe those fields which diverge or converge, e.g., the static electric field. The Divergence Theorem v ia dv = s Aids Lecture 1 Introduction & recap on 7 F. Rahman

8 Curl of a Vector Curl A = A A.dl = circulation of the vector A around a c closed contour A = Lim S 0 a n c A dl S max A non zero curl of a vector indicates the presence and measure of a vortex source or sink causing the maximum circulation per unit area. The curl of a field is conveniently used to describe those fields which loop, e.g., the static magnetic field. Stoke s Theorem When area S is very small, s ( ) whereby ( ) S A id s = A.dl c A ids = 0 Lecture 1 Introduction & recap on 8 F. Rahman

9 Examples of divergence and curl of EM fields: 1. Static electric field in a charge free region: i E = 0 and E = 0 2. Static Electric field: i E 0, E = 0 3. Static magnetic field: i B = 0, B 0 4. Electric field in a charged region with time varying magnetic field: Null Identities i E 0, B 0 1. ( V) = 0 Thus, if A = 0, then A = Φ where Φ is a scalar point function. 2. i A = 0 Thus if i B = 0, then B = A Helmholtz s Theorem A vector field is completely determined when its divergence and curl are specified everywhere. Lecture 1 Introduction & recap on 9 F. Rahman

10 See the two pages inside the back cover for some useful vector identities and formulae describing fields in Cartesian, cylindrical and spherical coordinates. Lecture 1 Introduction & recap on 10 F. Rahman