何曼君第三版高分子物理答案

22、试推导Debye色散方程式,并求出复介电常数的实部??与虚部???以及

??,tan?max 等特征值表达式。(对) ?max?''max??s???解:已知 ????11???K????????1????????1?i???????2???1 ?s?K?????????????????2?

??s?2????1?K????????????????3????2??4? 式中K??NA

3M联立以上三个方程式。 将(2)式减(3)式,得

?s?1???1??K??————(4) ?s?2???2将(3)、(4)式带入(1)式得:

???1???2?K???K??1?1?i???

??1?1s?? ???????2???2??2?1?i??????s???1????2??1?i????????1????2?????1????2???ss??s?

?????2???s?2??1?i?????s??1???1??i?????i????2?i????2i??????s????2???s?2??1?i?????s??2???s?

从而 ???1??????2???s?2??1?i?????????2?????si????LL?

何曼君第三版高分子物理答案

????i???????i????2???i????2??i?????????2????????2???ss?s?s??2??i????2?i????4?i????4i????2???4??2??4?ss?s?s???????2????2????4???????i????2???i????2???i????4??i????ss??ss?????2??2??4???i????2?i????2?i????4i???s?s??ss?3????6???3???i????6??i????3??i????6?i????3???6??s?s?s?s??2?i???????2?s???＝s??2i??????2s???2?s令?????2???2?即??????2s则

???????????????2???i????s????2????i????s1?i????????2?i?????????2??????i???????s1?i?????

?????s? ————(5) ?1?i?????这就就是Debye色散方程式。 将(5)式右边通分,并上下同乘?1?i???

??????1?i????1?i??????s?????1?i???

?1?i????1?i??? ????1??2?2???s??????i?????i??1???22

?s????2?2?s????????i ?22221???1????s?????s?????????i ????1??2?21??2?2何曼君第三版高分子物理答案

∴ ???????s???————(6)

1??2?2??????s??????————(7)

1??2?2?????s??????————(8) tg??????s??2?2????,应当令求?maxd????0 d??????????d?s??221???d???? ??d?d? ???s??????1??2?2????s????????2?2??1????222?0

∴ 1??2?2?2?2?2?0

?2?2?1 舍去负根

???∴ ?max???1 代入(7)式

?s???2

同样,要求tg?max,令

dtg??0 d???????????d?s2?2?????dtg??? ??sd?d????s???????s??2?2??????s???????????2?2???s??????222?0

∴ ?s??2?2???2???2?2?0 ?2?2?代入(8)式

?s?s ??? ?????s???s????s????? tg?max???s??????2?s??s?s?????

何曼君第三版高分子物理答案

∴ tg?max??s???2?1?s??

3、导出在交变电场中单位体积的介质损耗功率与电场频率的关系式,并讨论当ω→∞时介质的损耗情况。 解:Pr?Ir?U?cos0??Ir?U

又 Ir?????CoU,??????s??????

1??2?2∴ Pr2??????CoU????????CoU2???s

1??2?2Pr?s????Co?U2???2??

1??2?2当???时,1??2?2B?2?2 Pr???s????Co?U2

所以???时介质的损耗功率趋于定值。

4、假定某种高聚物的电导率为10-9欧姆-1米-1,载流子迁移率借用室温下烃类液体中离子载流子的数值10-9米2/伏·秒,计算高聚物的载流子浓度,并估算高聚物中重复单元的数量密度(假定重复单元相对分子质量为100,高聚物的密度为1),比较所得结果并加以讨论。 解:??n?q??

n???q???

已知电导率??10?9??1m?1

载流子迁移度??10?9m2V?s

每个载流子电量q?1.6022?10?19库仑?1.6022?10?19?A?s ∴n?10?910?9?1.6022?10?19m?3?6.2?1018m?3 假定高聚物密度为1,重复单元相对分子质量为100。 则每m3中含有的重复单元数为

106g6.023?1023n??3??6.023?1027m3

m100g可见每109个重复单元才出现一个载流子,载流子就是很少的,说明该高聚物

有很好的电绝缘性能。