??? Jsa?n?H?B?B?acoskz?acoskz r?azr?azra?A?A??sinkzr?a?sinkz ?sa?n?Dr?a?ra??在r?b的导体面上，法线n??ar，所以 ??? Jsb?n?H?B?B??acoskz??acoskz r?bzr?bzrb?A?A???n?Dr?b??sinkzr?b??sinkz
rb ?sb
???例5.6 已知真空中电场强度E?axE0cosk0(z?ct)?ayE0sink0(z?ct)， 式中k0?2??0??c。试求：
（1） 磁场强度和坡印廷矢量的瞬时值。
?（2） 对于给定的z值（例如z＝0），试确定E随时间变化的轨迹。
（3） 磁场能量密度，电场能量密度和坡印廷矢量的时间平均值。 解：（1）由麦克斯韦方程可得
???Ey??Ex??E??ax??ay?z?z?? ??axE0k0cosk0(z?ct)?ayE0k0sink0(z?ct)
??H???0?t对上式积分，得磁场强度瞬时值为
??E0?E H?axsink0(z?ct)?ay0coks0(z?ct)
?0c?0c故坡印廷矢量的瞬时值
2????E0 S?E?H??az
?0c?E（2）因为的模和幅角分别为
?Esink0(z?ct)22E?Ex?Ey?E0， ??tan0?k0(z?ct)
E0cosk0(z?ct)?所以，E随时间变化的轨迹是圆。
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（3）磁场能量密度，电场能量密度和坡印廷矢量的时间平均值分别为
??1?av,e?Re[E?D?]
4j(?k0z)?j(?k0z)1?????jk0zjk0z2 ?[(axE0e?ayE0e)?(ax?0E0e?ay?0E0e2)]
4?? ? ?av,m1?0E02 212??0E0 22?1????E0 Sav?Re[E?H]??az22?0c
例5.7 试将麦克斯韦方程组写成8个标量方程。
解：已知麦克斯韦方程组的积分形式为：
????????????D??B????lH?dl??S??J??t???dS,?lE?dl???S?t?dS,?SB?dS?0,?SD?dS?q
??????D,微分形式为：??H?J??t???B??E??,?t???B?0,???D??
?ax????Ax?Ay?Az又因为直角坐标系中 ??A?，??A????x?x?y?zAx?ay??yAy?az? ?zAz?ra????rA??az? ?zAz?ar?1??1?1?A??Az柱坐标系中??A?，??A?(rAr)??r?rr?rr???zAr?1?21??A1?A?球坐标系中??A?2(rAr)?， (sin??)?r?rrsin?????rsin????ar?1???A?2rsin??rAr?ra????rA??rsin?a?? ??rsin?A?(1) 直角坐标系中，麦克斯韦的积分方程可写为：
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??H??(a?dy?a?dz)??J?Dx???lxyz?S???dydzx?x?t???????lH??(a?dx?a?dz)???Dy??yxz?SJ??dxdz， y?y?t??????lH???(a?dy?a?dx)??Dz?zyx??S?Jz?z??t???dxdy??E??(a???Bxydy?azdz)????lx?S?dydzx?t???lE??(a???Byxdx?azdz)???Sdxdz ?yy?t???E??(a?dy?a?dx)??Bz??lzyx??S?dxdyz?t?SBdydz?xx?SBydxdz??Bzdxdy?0
ySz?SDxxdydz??SDydxdz??Dzdxdy?q
ySz麦克斯韦的微分方程可写为：
???Hz?Hy?Dx??Ez?Ey?By??J????zx?t???z??x?t???Hx??HzD??y?y??Ex?By??z?x?Jy??t， ???Ez??，??H?D??z?x?ty???Hx?J?z??Ey??Ex???Bz??x?yz?t???x?y?t(2)柱坐标系中，麦克斯韦的积分方程可写为：
???H?(a?rd??a?dz)??J??Dr???rd??l?rz?S?r?r?t??dz??H??(a?dr?a?dz)???J??D??????l?rz?S??t?drdz??????????lH?(adr?a?rd?)???J??Dz?zr??Sz?z?t???rdrd???E??(a??B?rd??a?rzdz)????lr?S?dzr?trd??????B?drdz ??lE?(ardr?azdz)????S?????lE??(a?dr?a?rd?)???Btzzr???Srdrd?z?t?SBrd?dz?rr?SB?drdz??Bzrdrd??0
?Sz38
?Bx?By?Bz?x??y??z?0?Dx?x??Dy?Dz?y??z??，
?SrDrrd?dz??D?drdz??Dzrdrd??q
S?Sz麦克斯韦的微分方程可写为：
?1?Hz?H??1?Ez?E??Dr?Br??J????r??r???z?t?z?t??r???D??B???Hr?Hz??Er?Ez， ?， ??J???????z?r?t?z?r?t???H?D1?1rz??1?(rE)?1?Er???Bz(rH?)??Jz???r?r?r?rr???tr???t??1?1?B??Bz1?1?D??Dz(rBr)???0， (rDr)???? r?rr???zr?rr???z(3)球坐标系中，麦克斯韦的积分方程可写为：
????Dr?2????lH?(a?rd??a?rsin?d?)??S?Jr???rsin?d?d?rr?t???????D????H?(adr?arsin?d?)?J??rsin?drd????lr??S????t???????D???????H?(adr?ard?)?J??rdrd?r????l??S????t???，
?Br2????E?(ard??arsin?d?)?????Sr?trsin?d?d???lr????B???E?(adr?arsin?d?)????lr??S??trsin?drd? ?????B???E?(adr?ard?)??rdrd?r????lS???t???SrBrr2sin?d?d???B?rsin?drd???B?rdrd??0
S?S?SrDrr2sin?d?d???D?rsin?drd???D?rdrd??q
S?S?麦克斯韦的微分方程可写为：
?H??E??Dr?Br???1?1[(sin?H)?]?J?[(sin?E)?]???r??rsin????rsin??????t???t???D??B??11?Hr??11?Er?， ， [?(rH)]?J?[?(rE)]????????t?t?rsin????r?rsin????r?D??B??Hr?Er?1??1?[(rH)?]?J?[(rE)?]??????????t???t?r?r?r?r?B?1?21?1?B?(rBr)?(sin?)??0，
rsin?????rsin???r2?r39
?D?1?21?1?D?(rDr)?(sin?)???
rsin?????rsin???r2?r
???9例5.8 已知在空气中E?ay0.1sin(10?x)cos(6??10t??z)(V/m)，求H和?。
?2???E解：由于电场强度E应满足空气中的波动方程?2E??0?02?0
?t2???Ey?29由于E?ay0.1sin(10?x)cos(6??10t??z)?ayEy，有?Ey??0?0?0 2?t
??Ey?2?2Ey?x2??2Ey?y2??2Ey?z2
??0.1(10?)2sin(10?x)cos(6??109t??z)?0?0.1?2sin(10?x)cos(6??109t??z)且
?2Ey?t2??0.1(6??109)2sin(10?x)cos(6??109t??z)
?2Ey?t2代入?Ey??0?02?0中，有?(10?)2??2??0?0(6??109)2?0
解得??10?3?54.41(rad/m)
???B又由麦克斯韦方程有??E??
?t???aaaxyz???B?????E??E?????E???axy?azy?t?x?y?z?z?x
0Ey(x,z)0???ax0.1?sin(10?x)sin(6??109t??z)?az?cos(10?x)cos(6??109t??z)????t?B?ax0.1?sin(10?x)cos(6??109t??z)?az?cos(10?x)sin(6??109t??z)B??dt????t6??109????B?a10?x)cos(6??109t??z)?az?cos(10?x)sin(6??109t??z)x0.1?sin( ?H??9?06??10???10?3?54.41rad/m，?0?4??10?7 ???H??ax2.3?10?4sin(10?x)cos(6??109t?54.41z) ??49?az1.3?10cos(10?x)sin(6??10t?54.41z)(A/m)40