电磁场与电磁波第四版习题解答

电磁场与电磁波(第四版)习题解答

第1章习题

习题1.1

,

给定三个矢量A、B和C如下

,

:A?ex?ey2?ez3.

B??ey4?ezC?ex5?ez2

解:

uuruurururuurAuruururex?2ey?3ez1u(1)aA?u?(ex?2ey?3ez) r?22214A1?2?(?3)ururuuruurur(2)A?B?ex?6ey?4ez?12?62?(?4)2?53 ururuuruururuurur(3)A?B?(ex?2ey?3ez)?(?4ey?ez)??11 ururA?B?11(4)??arccos?arccos?135.5?

AB1417??????A?BA?B11(5)AB?Acos?AB?A??????

17BB?Auuruururexeyezururuuruuruur(6)A?C?12?3??4ex?13eY?10eZ 50?2uuruururexeyezururuuruuruur(7)B?C?0?41?8ex?5eY?20eZ

50?2

urururuuruuruuruuruuruur A?(B?C)?(ex?2eY?3eZ)?(8ex?5eY?20eZ)??42

uuruururexeyezururuuruuruur A?B?12?3??10ex?eY?4eZ

0?41urururuuruuruuruuruur (A?B)?C?(?10ex?eY?4eZ)?(5ex?2eZ)??42

uuruururexeyezurururuuruuruur(8)(A?B)?C??10?1?4?2ex?40eY?5eZ

50?2uuruururexeyezurururuuruuruurA?(B?C)?12?3?55ex?44eY?11eZ

8520习题1.4给定两矢量 A?ex2?ey3?ez4和 B?ex4?ey5?ez6,求它们之间的夹角和 A在 B上的分量。

解:

?A?22?32?(?4)2?29 ?B?42?(?5)2?62?77

????????A?B?(ex2?ey3?ez4)?(ex4?ey5?ez6)??31

??则A与B之间的夹角为

?AB??A在B上的分量为

???arcis????????31?A?B???1310 ????arcos???A?B??29?77???????A?B?B?31AB?Acos?AB?A????A?????3.532

77BA?B

25, r2(1)求在直角坐标中点(?3,4,?5)处的E和Ex;

习题1.9用球坐标表示的场E?er(2)求在直角坐标中点(?3,4,?5)处E与矢量B?ex2?ey2?ez构成的夹角。 解:

(1)由已知条件得到,在点(-3,4,-5)处,

r?x2?y2?z2?(?3)2?42?(?5)2?52 ur2525E?2??0.5

r50?????e??2525r3?e4?exyz5E?er2?3?

rr102则 Ex?(2)其夹角为

?3102??32 20?EB

???arccos??????E?B??19?2?????153.6? ????arccosE?B??3?102??2A?er?ez2z验证散rr?5z?0z?4习题1.17在由、和围成的圆柱形区域,对矢量

度定理。

证:

在圆柱坐标系中

?1????A?(??2)?(2z)?3??2

????z?42?5所以, ???AdV??dz?d??(3??2)?d??1200?

V000又

????????A?dS??A?dS??A?dS?SA?dS??S上S下S柱面2?5?2?5?????A?ez?d?d????A00z?4040z?02?4???(?ez)?d?d????A00??5??e?5dzd?

??则

2?0?502?4?d?a???2?0?502?5dzd??1200???????AdV?1200???A?dS

VS22A?ex?ex?eyz沿xy平面上的一个边长为2的正方形回路的线xyz习题1.21求矢量

积分,此正方形的两边分别与x轴和y轴相重合。再求??A对此回路所包围的曲面积分,验证斯托克斯定理。

证:

??2??A?dl??AC020y?02???exdx??A022x?22???eydy??A02y?22???(?ex)dx??A0x?0??(?ey)dy

??xdx??22dy??xdx??0dy?8000

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