电磁场与电磁波总结
第1章 场论初步
一、矢量代数
A?B=ABcos?
A?B=eABABsin?
A?(B?C) = B?(C?A) = C?(A?B) A? (B?C) = B (A?C) – C?(A?B) 二、三种正交坐标系 1. 直角坐标系
矢量线元 dl?exx?eyy?ezz矢量面元 dS?exdxdy?eydzdx?ezdxdy 体积元 dV = dx dy dz
单位矢量的关系 ex?ey?ez ey?ez?ex ez?ex?ey 2. 圆柱形坐标系
矢量线元 dl?e?d??e??d??ezdzl 矢量面元 dS?e??d?dz?ez?d?d? 体积元 dV = ? d? d? dz 单位矢量的关系 e??e??ez3. 球坐标系
矢量线元 dl = erdr + e? rd? ? e? rsin? d? 矢量面元 dS = er r2sin? d? d? 体积元 dv = r2sin? dr d? d? 单位矢量的关系 er?e??e?
e??ez=e?ez?e??e?
e??e?=ere??er?e?
?Ar??cos?????A?????sin??A???z??0sin?0??Ax???cos?0? ??Ay?
?01????Az?cos???Ax????sin?? ??Ay?
?0????Az??Ar??sin?cos?sin?sin?????A????cos?cos?cos?sin??A???cos?????sin??Ar??sin?????A????cos??A??????00cos???Ar???0?sin?? ??A??
?10????Az?三、矢量场的散度和旋度
?A?dS1. 通量与散度???SA?dS divA???A?S?limv?0?v
2. 环流量与旋度???maxlA?dl rotA=elA?dln?lim?S?0?S
3. 计算公式
??A??Ax?Ay??x??y?Az?z ??A?1????(?A1?A??Az?)??????z ??A?1?21?1?A?r2?r(rAr)?rsin???(sin?A?)?rsin???
exeyeze??e?ezer re???A?????????x?y?z ??A??????z ??A??r ???AxAyAzA??A?AzAr rA?4. 矢量场的高斯定理与斯托克斯定理
?SA?dS??V??AdV
?lA?dl??S??A?dS
四、标量场的梯度 1. 方向导数与梯度
?uu(M)?u(M0)?l?limP?l?00?l
?u?l??u?u?P0?xcos???ycos??u?zcos? ?u?el??ucos? gradu??u?ne?u?u?un?ex?x?ey?y+ez?z 2. 计算公式
?u?e?ux?x?e?u?uy?y?ez?z ?u?e?u1?u????e?u?????ez?z ?u?e?u1?u1?ur?r?e?r???e?rsin??z 五、无散场与无旋场
1. 无散场 ??(??A)?0 F???A 2. 无旋场 ??(?u)?0 F??u
六、拉普拉斯运算算子 1. 直角坐标系
rsin?e????rsin?Az ?2u?2u?2u?u?2?2?2?x?y?z22222?2A?ex?2Ax?ey?2Ay?ez?2Az22222?2Ax?2. 圆柱坐标系
?Ay?Ay?Ay?Ax?Ax?Ax?Az?Az?Az22?? ， ?A??? ， ?A???2yz?x2?y2?z2?x2?y2?z2?x2?y2?z1???u?1?2u?2u?u??2????22???????????z2
??212?A??12?A??2?2A?e???2A??2A??2?e?A?A??????ez?Az?2?2????????????3. 球坐标系
1??2?u?1???u?1?2u?u?2?r ?sin?????r?r??r?r2sin???????r2sin2???22?A???222cot?2?A?2??2A?er??A?A?A??rr?2222?rrr??rsin???????22?Ar12cos??A??? ?e???A??A???22222??????rrsin?rsin????2?Ar212cos??A????e???A??A?????22222????rsin?rsin?rsin???七、亥姆霍兹定理
如果矢量场F在无限区域中处处是单值的，且其导数连续有界，则当矢量场的散度、旋度和边界条件（即矢量场在有限区域V’边界上的分布）给定后，该矢量场F唯一确定为
F(r)????(r)???A(r)
其中 ?(r)?14????F(r?)1? dVA(r)??Vr?r?4????F(r?)?Vr?r?dV?
第2章 电磁学基本规律
一、麦克斯韦方程组 1. 静电场基本规律
真空中方程：
? SE?dS?q4??0q?0
? lE?dl?0 ??E?? ??E?0 ?0场位关系：E(r)??r?r'r?r'V'?(r')dV' E???? ?(r)?314π?0??(r?)|r?r?| V?dV
介质中方程：
? S D?dS?q
? lE?dl?0 ??D?? ??E?0
极化：D??0E?P D?(1??e)?0E??r?0E??E 极化电荷：?PS?Pn?P?en ?P????P 2. 恒定电场基本规律
电磁场理论知识点总结
