Section 1 4 TRƯỜNG ĐIỆN TỪ

No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]

Introduction to Electromagnetic Fields,

to supplement “Elements of Engineering

Electromagnetics, Sixth Edition”

by

Nannapaneni Narayana Rao

Edward C Jordan Professor of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India

Scalar and Vector Fields

FIELD is a description of how a physical quantity varies from one point to another in the region of the field (and with time)

(a) Scalar fields

Ex: Depth of a lake, d(x, y)

Temperature in a room, T(x, y, z)

Depicted graphically by constant magnitude contours or surfaces

y

x

d1

d2 d3

(b) Vector Fields

Ex: Velocity of points on a rotating disk

v(x, y) = v x (x, y)a x + v y (x, y)a y

Force field in three dimensions

F(x, y, z) = F x (x, y, z)a x + F y (x, y, z)a y

+ F z (x, y, z)a z

Depicted graphically by constant magnitude contours or surfaces, and direction lines (or stream lines)

Example: Linear velocity vector field of points on a rotating disk

(c) Static Fields

Fields not varying with time

(d) Dynamic Fields

Fields varying with time

Ex: Temperature in a room, T(x, y, z; t)

D1.10 T(x, y, z, t)

(a)

Constant temperature surfaces are elliptic cylinders,

= To  x 1 sin  t   2 1 cos y  t   4z

0

0

, , , 0 1 0 2 1 1 4

4

        

x2  4z2  const.

Constant temperature surfaces are spheres,

(c)

Constant temperature surfaces are ellipsoids,

T x y z

0

0

, , , 1 1 0 2 1 1 4

16 4

        

x2  16y2  4z2  const.

Procedure for finding the Equation for the Direction Lines of a Vector Field

The field F is

tangential to the direction line at all points on a direction line.

dl F 

ax ay az

dx dy dz

F x F y Fz

0

dx

F x  dy F y  dz F z

dl F

F F

dl

dr

F r r d



F dz F z

dr

F r r d



F  r sin

F

cylindrical spherical

P1.26 (b)xa x  ya y  za z

(Position vector)

dx

x dy y dz z

ln x ln y  ln C1 ln z  ln C2

ln x ln C1y ln C2z

x C1y C2z

 Direction lines are straight lines emanating radially from the origin For the line passing through (1, 2, 3),

1 C1(2) C2(3)

C1 1

2, C2 13

x y

2 z3

or, 6x 3y 2z