均匀带电薄圆盘场强分布的研究

均匀带电薄圆盘场强分布的研究

黎印中

(贵州师范大学物理与电子科学学院 贵阳 550001)

摘要：通过对均匀带电细圆环空间电场的求解。在利用电场的叠加原理，导出均匀带电薄 盘在场点与源点的距离大于圆盘半径时，电场的级数表达式。 关键词：细圆环；电场；薄圆盘；叠加原理；级数表达式

Uniformly charged thin disc field distribution of

Abstract: Based on a uniformly charged ring solution of the electric field of space.The use of electric field superposition principle, derived the presence of a uniformly charged thin disc-point distance from the source point is larger than disc radius, the electric field of the series expression.

Key words: fine ring; electric field; thin disk; superposition principle; series expression

1 引言

在大学物理教材上，均匀带电薄圆盘作为一个典型的带电模型，常常需要求其空间的电场分布。本文借助文献[2]求出的均匀带电细圆环电场的空间分布，通过电场的叠加原理，导出均匀带电薄圆盘在以圆盘的中心为球心，以圆盘的半径为球半径的球外空间的电场分布。

2 均匀带电细圆环的电场分布的级数解

设圆环的半径为r0，电荷线密度为?。选择在球坐标系中，如图所示，由对称性可知，带电细圆环的电场分布关于z轴对称。因此，电场分布与?无关。为

了计算的方便，只求在xoz平面的任意一点的p处的电场，就可以代表整个空间的电场分布。

在xoz平面内任取一场点p（r，?，0），在圆环上任取一电荷元dq，源点dl(r0,各自位矢为

0rr10?2,?)

r?err，

r?er0r10。所以

r?? r?r ?er?er 。由球坐标基矢(er,e?,e?)与直角坐标基矢

(ex,ey,ez)之间的变换关系为[1]

ex?ersin??cos??eθcos??cos??e?sin?

ey?ersin??sin??e?cos??sin??e?cos?

ez?ercos??eθsin?

所以p点的ex，ey，ez分别为

?ersin??eθcos?

ey?e?

ez?ercos??eθcos?

又r0?r0cos?ex?r0sin?ey?r0cos?ersinθ?eθcosθ?r0sin?e??r0cos?sinθer?r0cos?cosθeθ?r0sin?e???

?r?? r?r0?err?rr1r0 ?err??0?r?rcos??sin??e?rcos??cos?e?rsin?er0?r0cos?sin?er?r0cos?cos?eθ?r0sin?e?0?θ?又在直角坐标系中p(rsinθ,0,rcosθ)，dl(r0cos?,r0sin?,0)。所以p,dl之间的距离| r?|=

?rsin??r0cos??2??r0sin??2??rcos??2

=r?r0?2rr0sin?cos? 令A= r2?r0

2?22?12

B?2rr0sin? 22r?r011所以| r?|=A2?1?BCOS??2,而电荷元dq=?r0d?，dq在p点产生的电场dE

?????r0?d??r??r0?d??r?dq?r? dE???334??0r?34??0r?34??0A2?1?Bcos??2?又dE在球坐标下的三个分量为

??r0??r?r0sin?cos???d?dEr??3 （1）

4??0A2?1?Bcos??32??r02cos?cos??d????dE?34??0A2?1?Bcos???r02sin??d?4??0A?1?Bcos??323232 （2）

????dE?

（3）

对<1>、<2>、<3>积分得

2?r?rsin?cos??r00??Er?d? （4） 3?30?1?Bcos??24??A20???E??r02cos?4??0A32?02?cos?0?1?Bcos??sin?32?d? （5）

???E??r024??0A32?2??1?Bcos??32?d? （6）

因为

?2?0?0cons??d????2?Cnn为正奇数n为正偶数

其中Cn?(n?1)(n?3)(n?5)???3?1

n(n?2)(n?4)(n?6)???4?22rr0sin?r?r0?32因为B?22?1，所以?1?Bcos??可作泰勒展开

?3235?37?5?323n???????1?Bcos??Bcos??Bcos??????DBcos?????n22!?223!?23(2n?1)(2n?1)???3?1Dn?

n!?2n所以

?1?Bcos?????E????r02cos?324??0A2?r0cos?2?32?2?cos?0?1?Bcos??32?d?4??0A?r02cos??317?5?333?1?k??B??B?????DBC????2?kk?13?3?224?23!?2??4??0A2?r02cos??????DkBkCk?13?05?3?3?2ncos??1?Bcos??(Bcos?)?????D(Bcos?)????d?n2?22!?2??

??4??0A2k?1,3,5Er???r0324??0A332??r0?2????2d??2d??????r?1?Bcos?rsin?cos?1?Bcos?3??0?00???4??0A22???r035?3?2n?r1?(Bcos?)?(Bcos?)?????D(Bcos?)????d?n3?0?2?2!?2?2?4??0A22??r05?3?3?2n?rsin?cos?1?(Bcos?)?(Bcos?)?????D(Bcos?)????d?0n32?0??2!?2?2?4??0A2?r0?5?321?n??r?1??B??????DBCnn?2?32?2?2!?2?4??0A22???r035?32?23nn?1?rsin?cos??Bcos???B?cos??????DBcos?????d?n302?0??22!?2??4??0A2???r0???317?5?333?1??nk??r?1??(DnBCn)?2???r0sin??B??B?????DkBCk?1?????2??3??3224?23!?2???n?2,4,6???4??0A2??????r0?nk?2?r?2?r(DBC)?rsin??2?DBC??nn0kk?1?3? n?2,4,6k?1,3,5?2?4??0A????r0?nk?r?r?(DnBCn)?r0sin??DkBCk?1?3?n?2,4,6k?1,3,5?2?0A2??2?0??rsin?cos?r0???d??33??1?Bcos??2?1?Bcos??2??????? 2 均匀带电薄圆盘的电场分布

又均匀带电薄圆盘是由无数均匀带电细圆环组成，用电场叠加原理，球出组成薄圆盘的所在细圆环在P点处的场强，就可代表整个圆盘在空间中的电场分布了。

设均匀带电薄圆盘的电荷面密度为σ，圆盘的半径为a,距圆心r0处，宽度为

dr0的细圆环的电荷线密度为λ=?dr0，于是P点的电场为

?5?321nC)?rsin?(3B?1?7?5?3B33?1?????DBkC)??drr?r(?B??????DB?0nn0kk?13?232224?2??2!?23!?2??2?(r2?r2)200?rr0?rr0?aan??0?dr0??0?(DnBCn)?dr02323n?2,4,6rr2?0r3?(1?0)22?0r3?(1?0)2r2r2?r0?a??0?r0sin??DBkC?dr0k?1,3,5kk?123222?0(r?r0)aEr??0?r0??r又02?1时

r2222r0?3rrr33?53?5???(2j?1)jj(1?)2?1?(?1)1?0?(?1)2(0)2?????(?1)(0)

2r2j!?2r22!?22r2r22222???rrrr33?502j3?5???(2j?1)0j?aa?dr0??00?1?(?1)1?0?(?1)2()?????(?1)()dr0?0232222?2rj!?22?0r?2!?2rr2?r???2?0r3?(1?0)2r2?2?462j?2rraa??r0a3r0a3?53?5?????(2j?1)j?200??(?1)???(?1)?????(?1)??????jj2r2?402?0r2?202!?22(r2)2?60j!?2(r2)?(2j?2)0????rr02j?2?aj3?5?????(2j?1)??(?1)jj22?0rj?0,1,2,???j!?2(r2)?(2j?2)?又

?n?(DnBCn)?dr0?23n?2,4,6r2?0r3?(1?0)2r2?rr0?aDnBnCn?dr0??023n?2,4,6,???r2?0r3?(1?0)2r2?rr0DnCn2rr0sin?n?a?()?dr0??0222n?2,4,6,???r03r?r032?0r?(1?)2r2nrn?1sinn??rn?1?DC2?ann0??dr0??02n?2,4,6,???r0n?32n?322?0r?(1?)r2a?0?2n?1?DnCnsinn??r?n?2a???0?n?2,4,6???0n?1r0

?dr02r0n?32(1?)r2?rr0r0?1时，有 2r2

本文来源：https://www.bwwdw.com/article/g7pf.html