Chapter 19 Maxwells Equations The Total Current Density J a The curl of b ■ Maxwells Equations Maxwell' s equations in integral Form
Chapter 19 Maxwell’s Equations ◼ The Total Current Density J ◼ The Curl of B ◼ Maxwell’s Equations ◼ Maxwell’s Equations in Integral Form
So far we have obtained maxwells four equations V·E=f+ V.B=0 OB V×E VXB=uo(Jf +Je);(steady G enerally
19. 1 The Total Current Density J There are three kinds of current densities as follows 1. The free current density J1 f The displacement current density OD a OP ot Eoe+ at 3. the equivalent current density in magnetic ma terial V×M
Thus the total current density is the sum a OP f + E0 E++V×M ot at =Jn+609t where J m is the volume current density in matter OP f Ot+V×M
19.2 The Curl of b The fourth Maxwell equation VXB-uoJf +Je);(steady) should be generalized with the currents Jf+Je re- placed by the total current Jt V×B=p40Jt OP 10(J+60aE++V×M) 0 (Jm+0。E) at or. rewritten as V×B-10∈0aE=p0Jm
ugging B=10(H+M), Into a OP V×B=0(Jf+∈0。E+ a+V×M we get a OP V×H=Jf+0。E+ ot J OD ot
Example: Dielectric-Filled Parallel-Plate Capacitor It is connected across a alternating source v and contains a slightly conducting dielectric with a per- mittivity erco and a conductivity o.(neglecting the side effects D Figure 19-1 Parallel-plate capacitor connected to a source of alternating voltage The current J,+aD/dr gives an azimuthal magnetic field B
What is the magnetic induction inside the capacitor? B=? Remember that the material inside is dielectric but not magnetic, so M=0, and B=0(H+M)=0H, therefore the maxwell equation OD Y×H=Jf+ at is simplified as OD V×B=10(Jf+ at Integrating this over an area s
we get OD /sVXB.da=0/(Jf+)·da Using Stokes Theorem, this is written as OD 允B·d=A0(Jf+) da ot where C is the curve bounding the surface et c to be a circle of radius r. Note that Je+ oD is homogeneous. So aD 27B=10(J+)4) nr at B=②(y+ aD at
Now Jf =oe=ov/s aD OE at - Ereo d' By the alternating property E=E(t)-EeJwt we have OD at erE0jWE-ercojwV/s, an B=(a+6r60j) S Thus although Jf+aD/at is independent of r, the magnetic field b depends on r and is azimuthal
