Chapter 19 Maxwell,s Equations The Total Current density J The curl of b Maxwell's 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 Maxwell s four equations + V·E V·B=0: OB VXE at VB=oJf + Je);(steady Generally?
19. 1 The Total Current Density J There are three kinds of current densities as follows The free current density Jf 2. The displacement current density OD 00P =(0E+ ot at at 3. The equivalent current density in magnetic ma terial VⅹM
Thus the total current density is the sum t=Je+(0ob⊥OP 0 +V×M 0t"0t 0 E Jm+ at where Jm is the volume current density in matter m=J× +V×M at
19.2 The Curl of b The fourth Maxwell equation V×B=0{Jr+Je);( stead should be generalized with the currents Jf Jere placed by the total current ji V×B a oP =0(Jf+0aE+ at +VM) 0 Jm+c0e or. rewritten as V×B-00aE=10Jm
Plugging B=0(H+M) Into 00P V×B=10(Jf+60。E+ +V×M) t at we get 00P V×H=Jf+60aE+ t at J D 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 Er 0 and a conductivity o.(neglecting the side effects) J aD or 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=10(H+M)=10H, therefore the maxwell equation VXH=动 01 is simplified as V×B=10(Jf+ OD at Integrating this over an area s
we get OD sV×Bda=0/s(J+) Ot′ da Using Stokes Theorem, this is written as OD CB·dl=p0/s(J+x)·da, where C is the curve bounding the surface Let c to be a circle of radius r. Note that Jf+ ot is homogeneous. So OD 2丌B= 丌T OD B=OUf+ a ) ot
Now E=oV/s aD OE By the alternating property E-E(t=Ee! we have OD at CrEo, WE=ErE]wVs and B=(a+60)Vr Thus although Jf+aD/at is independent of r, the magnetic field b depends on r and is azimuthal
