Section 3 2 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[.]

Slide Presentations for ECE 329, 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 3.2 Gauss’ Laws and the Continuity Equation 3.2-3 GAUSS’ LAW FOR THE ELECTRIC FIELD S D • dS  V  dv z (x, y, z) x  Dx x x  y  z   Dx x  y  z   Dy   z  x   Dy   z  x y  y y   Dz z  z  x  y   Dz z  x  y   x  y  z z y y x 3.2-4  D   x x x   Dx x  y  z    Dy    Dy  Δ z Δ x y +Δy y Lim x  y  z    Dz z z   Dz z   x y  x y  z   x y  z Lim x   x y  z y  z  3.2-5 Dx Dy Dz    x y  z       Longitudinal derivatives of the components of D   • D  Divergence of D =  Ex Given that 0 for – a  x  a    0 otherwise Find D everywhere 3.2-6 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •  • • • • • • • • • x=–a x=0 x=a Noting that  = (x) and hence D = D(x), we set    and  0, so that y z Dx Dy Dz Dx • D     x y z x 3.2-7 Thus, • D =  gives Dx   (x) x which also means that D has only an xcomponent Proceeding further, we have x Dx  –  x  dx  C where C is the constant of integration Evaluating the integral graphically, we have the following: 3.2-8  –a a x x –   ( x ) dx 0 a –a 0 a x From symmetry considerations, the fields on the two sides of the charge distribution must be equal in magnitude and opposite in direction Hence, C = – 0a 3.2-9 Dx 0 a –a a x – 0a – 0 a a x  D  0 x a x  0 a a x for x  –a for – a  x  a for x  a 3.2-10 GAUSS’ LAW FOR THE MAGNETIC FIELD  D • dS =   dv S From analogy V • D  S B • dS = = V dv  • B 0  • B 0 Solenoidal property of magnetic field lines Provides test for physical realizability of a given vector field as a magnetic field 3.2-11 LAW OF CONSERVATION OF CHARGE d  dv 0 J • dS  S dt V  • J  t (  ) 0    • J  t 0  Continuity Equation 3.2-12 SUMMARY B  E – t D  H J  t  • D  (1) (2) (3)  • B 0 (4)  • J  0 t (5) (4) is, however, not independent of (1), and (3) can be derived from (2) with the aid of (5)