电磁场与电磁波（杨儒贵第二版）课后答案-1

??以及sinac，cos??，求得 rr22c??a??a?c0?Ar??rr??a??r??r????a???????c??A?0?b?0????0? ?????rr?b??b??A???010???c??????????????????即该矢量在球坐标下的表达式为A?rer?be?。

1-22 已知圆球坐标系中矢量A?aer?be??ce?，式中a, b, c均为常数，A是常矢量吗？试求??A及??A，以及A在直角坐标系及圆柱坐标系中的表示式。

解 因为虽然a, b, c均为常数，但是单位矢量er，e?，e?均为变矢，所以A不是常矢量。

在球坐标系中，矢量A的散度为

1?21??sin?A???1??A?2rAr?r?rrsin???rsin?????A????????? ?将矢量A的各个分量代入，求得??A?矢量A的旋度为

2ab?cot?。 rrerr2sin????A??rAre?rsin????rA?e?r? ??rsin?A?err2sin????rae?rsin????rbe?r?b?e? ??rrsin?c利用矢量A在直角坐标系和球坐标系中各个坐标分量之间的转换关系

?Ax??sin?cos?????Ay???sin?sin??A???z??cos?cos?cos?cos?sin??sin??sin???Ar???cos????A??

?0????A??2222?x?x?yx?y??2??sin???cos2222ax?y?x?y?z?以及?，?，求得该矢量在直角坐标下的表达式为

yzz?cos???sin????222?ax2?y2x?y?z?? 13

A???bxzcy?cx??x???byz??ax2?y2?x2?y2?ex??y????ax2?y2?x2?y2?ey?

22 ???bx?y??z??e?a?z?利用矢量A在圆柱坐标系和球坐标系中各个坐标分量之间的转换关系

?z?Ar??sin?cos?0???ra0??b???a??r?az??A??????001??Ar????a01?????Az????cos??sin?0??A???0????A?????z??a?r??b????c?? a0?????c??????z?bar??求得其在圆柱坐标下的表达式为

A????r?baz???e?b?r?ce????z?ar??ez。

1-23 若标量函数?1(x,y,z)?xy2z，?2(x,?,z)?rzsin?，?3(r,?,?)?sin?r2，试求?2?1，?2?3。

解 ?2??2?1?2?1?2?11??x2??y2??z2?0?2xz?0?2xz ?2?1????2?1??2?2r?r??r?r?????2?22?r2?????2?????z2 ?1?r?r?rzsin???1r2??rzsin???0 ?2?1??2??3?1????3?1??23?r2?r??r?r???r2sin?????sin??????r2sin2???3?????2??? ?1??r2?r??r2?2sin??r3???1??sin?cos??r2sin?????r2???0 ?2sin?r4?cos2??sin2?1r4sin??r4sin? 1-24 若 A(x,y,z)?xy2z3ex?x3zey?x2y2ez A(r,?,z)?e2rrcos??ezr3sin?

A(r,?,?)?errsin??e1?rsin??e1?r2cos? 试求??A，??A及?2A。

?2?2及14

解 ①??A??Ax??Ay??Az?y2?x?y?zz3?0?0?y2z3； exeyezexeyez??A????????x?y?z??x?y?z AxAyAzxy2z3x3zx2y2??2x2y?x3?ex??3xy2z2?2xy2?e?2y?3xz?2xyz3?ez； ?2A?ex?2Ax?e2y?Ay?ez?2Az

??2xz3?6xy2z?ex?6xze?2y?2y2?2x?ez；

② ??A?1??rA?A?r?rr??1r????Az?z?1?r?r?r3cos???0?3rcos?

ereee?zrezr?re???r?r??A???r???z???r???z ArrA?Azr2cos?0r2sin??err?r2cos???e??ez???2rsin?r?r2sin?? ?errcos??2e?rsin??ezrsin??2A?e?????2AAr2?A???2A?2?Ar?rr?r2?r2??????e?????A??r2?r2??????e2z?Az ?2ercos??2e?sin??3ezsin?；（此处利用了习题26中的公式） ③ ??A?1?r2?r?r2Ar??1??rsin?sin?A???1??A???rsin?????????? ?1?r2?r?r3sin???1?rsin????r?1sin2???0 ?3sin??2cos?r2；

ere?e?ere?e?r2sinrsinrsin??A?????r?r2sin????r??r??????r????

ArrA?rsin?A?rsin?sin?r?1sin?cos? ?e?sin???2cos???sin??r???r3???e???r3???e????r2?cos???

15

??esin?r3?e2cos?r?r3?e?sin?????cos??r2??； ?2A?e?22??Ar???2Ar?r2Ar?r2sin????sin?A2?????r2sin????? ?e?2AA?2?Ar2cos??A????????r2sin2??r2???r2sin2????? ?e??2A?2?Ar2cos??A?????A??r2sin2??r2sin????r2sin2????? 将矢量A的各个坐标分量代入上式，求得

?2A?e??cos2?4cos???2cos?2sin??cos?r?rsin??r3???e???r?r3???e?r4sin2? 1-25 若矢量A?ecos2?rr3, 1?r?2，试求? V ??AdV，式中V为A所在的区域。解 在球坐标系中，dV?r2sin?drd?d?，

??A?1?21?r2?r?rAr???sin?A?rsin??????1rsin???A????????? 将矢量A的坐标分量代入，求得

??cos2V??AdV??V?????2??2cos2?2??r4??dV???0d??0d??1r4rsin?dr

???2??cos2?2?0d??02sin?d????0cos2?d????

1-26 试求

? S(er3sin?)?dS，式中S为球心位于原点，半径为5的球面。

解 利用高斯定理，?SA?dS??V??AdV，则

?A?dS????AdV??2??56sin?SV0d??0d??0rr2sin?dr?75?2 16