变速旋转带电体的磁偶极辐射阻尼力矩公式与算例

变速旋转带电体的磁偶极辐射阻尼力矩公式与算例 The Damping Torque Formula and Its Examples of Magnetic-Dipole Radiation for Rotational Charged Bodies with Changing Angular Velocity first-author 周国全，男，副教授，博士，现从事理论物理教学及场论与非线性可积方程研究。E-mail:zgq@whu.edu.cn 周国全，男，副教授，博士，现从事理论物理教学及场论与非线性可积方程研究。E-mail: zgq@whu.edu.cn zgq@whu.edu.cn 武汉大学 物理科学与技术学院, 湖北 武汉 430072 武汉大学 物理科学与技术学院, 湖北 武汉 430072 School of Physics and Technology, Wuhan University, Wuhan 430072, Hubei, China School of Physics and Technology, Wuhan University, Wuhan 430072, Hubei, China 通过电磁场矢势的远场多极矩展开理论，引入旋转带电体的一个本体电矩张量以计算其磁矩，推导了定轴变角速转动带电体的磁偶极辐射功率及其受到的辐射阻尼力矩公式。研究表明, 忽略延迟效应，在非相对论远场情形以及准周期平均意义下，变速旋转带电体的辐射阻尼力矩与其角速度的四阶导数有关；给出了若干特殊形状变速旋转带电体的磁距、磁偶极辐射功率和辐射阻尼力矩的具体算例；另外，以带电简谐转子为例, 利用微扰法讨论了辐射阻尼力矩对其转动的影响。 Deriving from the multipole-moment and far-field expansion theory of the electromagnetic vector potential, an electric quadrupole moment tensor was introduced to calculate the magnetic moment of a rotational charged body. And its magnetic-dipole radiation power was concretely derived, subsequently, the radiation damping torque upon the rotational charged body about a fixed axis was also deduced. The research results show that, if the retarded effect is ignored, the radiation damping torque is generated from the fourth-order derivative of the angular speed with respect to time, under the non-relativistic far-field case, and on average in a quasi-periodic process. Some calculated examples of magnetic moment, magnetic-dipole radiation power and the radiation damping torque, for several concrete charged bodies of special shape, were numerated and listed in a table. Moreover, a concrete rotational vibrator in a simple harmonic oscillation was taken as a reference to illustrate the general radiation theory. By means of perturbation method, the influence of the damping torque upon the rotational vibrator was also discussed in detail. 旋转带电体 电矩张量 磁偶极辐射 辐射功率 辐射阻尼力矩 rotational charged body charge moment tensor magnetic-dipole radiation radiation power radiation damping torque 引言 1 关于旋转带电体磁矩的计算及其转动规律的研究，以及变速旋转带电体的辐射功率和阻尼力矩的理论探索, 在电磁场理论及其应用领域具有特殊而重要的意义 [ 1 1 , 2 2 ] 。由于电荷变速运动时会产生电磁辐射，带电体变速转动时也会产生电磁辐射，变速直线运动的电荷会因辐射而受到阻尼力 [ 1 1 ] ，因此带电体变速转动时也会因辐射而受到阻尼力矩。对于旋转带电体的磁矩及其动力学规律, 本课题组已有一定的研究基础。文献[ 2 2 ~ 5 5 2~5 ]通过引入诸如电矩张量、标量电矩、磁矩椭球、主轴及主轴标量电矩等一系列概念，推导出任意带电体绕定点或定轴转动磁矩的若干规律；文献[ 5 5 ]明确给出了任意给定电荷分布的旋转带电体的零磁矩条件；文献[ 6 6 ~ 9 9 6~9 ]推导了旋转带电体磁矩的(推广的)平行轴定理与正交轴定理、中心轴定理等若干重要的计算法则及其算例。文献[ 10 10 , 11 11 ]基于旋转带电体磁矩的张量表达，利用经典力学中转动刚体的欧拉方程，推导了匀强磁场中定点转动带电刚体的拉格朗日动力学理论和欧拉方程，解决了两种简单对称情形下欧拉方程解的问题。本文将在此理论基础上, 利用已有的研究成果, 计算变速旋转带电体的磁偶极辐射功率及其辐射阻尼力矩等重要物理量, 并举例说明。 在非相对论情况和远场条件下，从运动带电系统电磁矢势的多极矩展开理论出发，一般教科书忽略其辐射场的高阶无穷小项，而仅考虑电偶极、磁偶极与电四极辐射。电偶极辐射及其阻尼力的理论已然成熟 [ 1 1 , 12 12 ~ 14 14 12~14 ] ，比其更高阶的磁偶极、电四极辐射场一般具有同一量级，并已有初步结论 [ 1 1 , 13 13 , 14 14 ] 。而对于变速旋转带电体，与电四极辐射相比，其磁偶极辐射的影响更为显著，但其辐射功率和阻尼力矩的具体计算，目前较少见诸文献[ 2 2 , 13 13 , 14 14 ]，此问题正是本文的研究重点。当带电体做定轴转动时，其磁矩、角动量、辐射阻尼力矩均与角速度在同一方向。本文借鉴文献[ 1 1 , 13 13 , 14 14 ]中推导变速运动电荷辐射阻尼力的方法与技巧，对于经历一个假设的周期性定轴转动的带电刚体，依据相关磁偶极辐射过程中能量守恒的原则，忽略延迟效应，并在非相对论远场情形以及准周期平均意义下，推导定轴变速转动带电体磁偶极辐射的阻尼力矩和功率表达式，并应用于带电转子的简谐转动过程。作为例子, 利用微扰法求出辐射阻尼力矩对转动振子转动状态的影响。 转动带电体磁矩的基本概念 1 关于旋转带电体的定点或定轴磁矩, 本课题组已通过两种方法, 引入了一个电荷矩张量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle ( O ), 得到其磁矩的张量积表达形式 [ 2 2 ~ 5 5 2~5 , 10 10 , 11 11 ] 。 设 J ( r ′)为源点 r ′处的电流密度, ρ e ( r ′)为该处电荷体密度, 则稳恒电流 J ( r ′)= ρ e ( r ′) v ′在远场点 r 处磁矩 P m 的计算公式是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257095&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257095&type=middle 对于以任意角速度 ω = ω ( t )作定点转动的刚体, 将 v ′= ω × r ′代入(1)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257101&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257101&type=middle 其中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257197&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257197&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257197&type=middle 为三维二阶单位张量矩阵， r ′ r ′是一个三维二阶并矢张量。由此，可定义一个三维二阶对称张量(相对给定点 O ) http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle ( O ) http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257108&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257108&type=middle 其中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle ( O )的张量元素可表示为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257112&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257112&type=middle 其中， r ′ 2 = x ′ 1 2 + x ′ 2 2 + x ′ 3 2 ，d 3 r ′=d x ′ 1 d x ′ 2 d x ′ 3 , δ αβ 为Kronecker记号， α =1, 2, 3, β =1, 2, 3。 因此, 由(2)式和(3)式可得旋转带电体相对于给定点 O 的磁矩 P m 的一个张量积表达公式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257114&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257114&type=middle 而上述张量与本课题组在文献[ 2 2 ~ 4 4 2~4 ]中，通过旋转带电体的磁矩与刚体的转动惯量之间的类比关系，引入的张量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle 完全一致。其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle ( O )定义于带电刚体的本体坐标系中, 不同于电动力学中既有的电四极矩, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle ( O )在由其特征向量所张成的主轴坐标系 Oxyz 中呈对角形式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257218&type=middle ( O )=diag( T 1 , T 2 , T 3 )。于是，(5)式可写为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257119&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257119&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257119&type=middle 其中， i , j , k 分别是 x , y , z 轴的单位矢量。 对于过 O 点，沿角速度矢量 ω = ω ( t ) http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=middle 方向作定轴转动的带电刚体，其标量定轴电矩与矢量定轴磁矩分别为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257123&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257123&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257123&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257127&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257127&type=middle 其中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=middle =(cos θ 1 , cos θ 2 , cos θ 3 )为 ω 方向单位矢量， θ 1 , θ 2 , θ 3 分别为角速度矢量 ω 与 x 轴、 y 轴、 z 轴的夹角。 定轴变速旋转带电体的磁偶极辐射功率及其阻尼力矩 1 将旋转带电体看成置于 O 点的磁偶极子，取磁距 P m 方向为极坐标的 z 轴，在非相对论远场情形并忽略延迟效应的情况下，在以 r 0 为半径的球面上，磁偶极辐射的能流密度公式为 [ 1 1 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257130&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257130&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257130&type=middle 其中, μ 0 为真空磁导率, c为真空中的光速, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257332&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257332&type=middle m 为磁矩 P m 对时间变量 t 的二阶导数， n 为径向(辐射方向)单位矢量, θ 为 P m 与 n 的夹角。对辐射方位角( θ ， φ )在 θ ∈[0, π]， φ ∈[0, 2π)范围内积分(带权重因子 r 0 2 sin θ )，得到磁偶极辐射功率 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257134&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257134&type=middle 其中，d W m 为带电体在d t 时间内辐射的能量。 假设，带电体(一般为刚体)对于绕定轴作变速转动的磁偶极辐射的阻尼力矩为 M ( M = M http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=middle ), W dp 为阻尼力矩的功, 在带电体绕定轴发生无穷小角位移d ϕ 时，由变力矩的元功公式 [ 15 15 ] : d W dp = M d ϕ , 可得阻尼力矩 M 的瞬时功率 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257136&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257136&type=middle 其中，瞬时角速度 ω = ω ( t ) http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257231&type=middle ，瞬时角速率 ω ( t )=d ϕ /d t 为矢量 ω 的大小。在此仅考虑磁偶极辐射，并采用标量定轴电矩 T l 的概念, 在远场区域, 根据(8)式, 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257139&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257139&type=middle 本文中， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257234&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257234&type=middle 、 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257236&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257236&type=middle 、 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257238&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257238&type=middle 分别表示 ω 对时间变量 t 的一阶、二阶和三阶导数，其他情形，以此类推。于是，磁偶极辐射功率为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257141&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257141&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257141&type=middle 根据能量守恒定律，辐射阻尼力矩的瞬时功率d W dp /d t 应等于旋转带电体瞬时辐射功率的负值与其固有电磁能量 E sf 的时间变化率之和，即有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257143&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257143&type=middle 将(11)式和(13)式代入(14)式, 可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257145&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257145&type=middle 考虑一个变速转动带电体，其角速率 ω 为时间的周期函数，在一个周期 T 内，带电体周围的自能部分 E sf 恢复原值，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257147&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257147&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257147&type=middle 所以 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257149&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257149&type=middle 若 ω 是周期函数，那么 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257243&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257243&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257243&type=middle 、 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257246&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257246&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257249&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257249&type=middle 也是周期为 T 的周期函数, 下式中的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257251&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257251&type=middle ，则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257151&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257151&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257151&type=middle 又由于 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257152&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257152&type=middle 且因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257253&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257253&type=middle , http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257257&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257257&type=middle , 故由(18)式和(19)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257260&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257260&type=middle , 代入(17)式等号右边，得到 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257153&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257153&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257153&type=middle 亦即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257154&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257154&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257154&type=middle 等式(21)对任意周期性的转动过程都须成立, 则下式必然成立，即定轴变速旋转带电体的辐射阻尼力矩表达式为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257155&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257155&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257155&type=middle (22) 式表明, 只有当带电体的角速度具有非零的四阶导数时, 才存在辐射阻尼力矩。 需要注意的是，推导过程是基于定轴转动时d W = M ·d θ = M d θ 的事实。这是因为对于定轴转动, 矢量平行关系 M //d θ 成立, 因而d W = M ·d θ = M d θ ; 对于定点转动, 平行关系 M //d θ 一般不再成立, 导致 M ·d θ = M d θ 也无法成立 [ 15 15 ] ，因此，(22)式的阻尼力矩公式不再适用。 当然, 类似于文献[ 1 1 , 12 12 ~ 14 14 12~14 ]中关于加速运动的带电粒子所受阻尼力的讨论, (22)式也不是严格成立的。它是在低速转动情形, 忽略远场区电磁场延迟效应, 并在固有场与辐射场达到相对稳定的准周期状态时, 才成立的公式。应该说它代表变角速定轴转动的带电体在其辐射场达到相对稳定的准周期状态时, 其在一个周期内所受到的来自于磁偶极辐射的平均阻尼力矩。 磁偶极辐射阻尼力矩对定轴转动带电体振子的影响 1 考虑一个在弹性回复力矩作用下，做周期转动的带电体转动振子。例如一个变角速定轴转动的带电体(定轴转动惯量为 J ，扭力矩系数为 C )，最初从静止开始，在线性的扭转回复力矩- Cθ 作用下，按简谐规律 θ ( t )= Θ 0 sin( Ω 0 t )， ω ( t )= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=middle = ω 0 cos( Ω 0 t )而转动, Θ 0 和 ω 0 (= Θ 0 Ω 0 )分别为 θ 和 ω 的最初振幅, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=middle 为角位移大小( θ = θ ( t ))对时间的导数， Ω 0 为带电体最初的振动圆频率。本文研究变角速定轴转动带电体的转动是如何受其磁偶极辐射阻尼力矩影响的。假设其电偶极矩与电四极矩皆为零, 只有磁偶极矩, 其矢量定轴磁矩 P m 与标量定轴电矩 T l 的关系仍如(8)式所示，即为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257156&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257156&type=middle 由于 ω ( t )= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=middle ( t )，结合(22)式可知，磁偶极辐射阻尼力矩为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257159&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257159&type=middle 其中， k ≡ μ 0 T l 2 /24πc 3 。且由(24)式可知， M = http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257269&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257269&type=middle 。根据刚体的定轴转动定律，转动振子的转动微分方程为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257162&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257162&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257162&type=middle 令 C / J = Ω 0 2 , 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257273&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257273&type=middle , (25)式即可改写为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257166&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257166&type=middle 计算磁偶极辐射阻尼力矩的影响时, 由于辐射阻尼力矩远小于弹性回复力矩, 可视之为微扰力矩，因此，无须求解非线性振动方程，而是采用微扰方法。先忽略辐射阻尼力矩，得到零阶近似解 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257168&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257168&type=middle 其解为简谐转动 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257172&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257172&type=middle 由(28)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257174&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257174&type=middle 再在转动微分方程((25)式)中加入阻尼微扰力矩((24)式)，并在阻尼微扰项中应用零阶近似关系 θ ( t )≈θ (0) ( t )和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=middle ( t )≈ http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257280&type=middle (0) ( t )，于是 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257177&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257177&type=middle 将(29)式代入(26)式右侧微扰项中，转动微分方程可改写为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257179&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257179&type=middle 由(26)、(29)和(30)式可知, (30)式中最后一个等式右边的系数 γ ≡ kΩ 0 4 / J , 称为转动阻尼因子。上述常微分方程(30)的特征根 λ 满足方程 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257182&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257182&type=middle 在微扰情形，即 γ ≪ Ω 0 时，有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257185&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257185&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257185&type=middle 考虑到转动的初始条件 θ (0)=0，方程(30)的解为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257188&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257188&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257188&type=middle 由(33)式可知，在仅考虑一阶微扰的情况下，由于转动阻尼因子 γ 较小，衰减时间常数 τ =2/ γ =2 J / kΩ 0 4 较大，因此，磁偶极辐射阻尼力矩将使振子的转动振幅 Θ 0 exp(- γt /2)按指数规律缓慢衰减，而简谐转动的圆频率 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257191&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257191&type=middle 与 Ω 0 相比略有减小。 特殊形状变速旋转带电体的磁矩、磁偶极辐射功率与辐射阻尼力矩 1 根据前述理论和计算公式, 对于一些具有均匀体(面)分布的带电刚体, 具体计算了以 z 轴为转轴时它们的磁矩、磁偶极辐射功率和辐射阻尼力矩，见 表 1 表 1 。 序号 旋转带电体形状 磁矩 P m 磁偶极辐射阻尼力矩 M 磁偶极辐射功率d W m /d t 图示 1 细直杆(线分布) Ql 2 ω /24 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257282&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257282&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257284&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257284&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257288&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257288&type=middle 2 矩形板(面分布) Ql 2 ω /6 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257290&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257290&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257290&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257294&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257294&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257294&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257296&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257296&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257296&type=middle 3 圆形板(面分布) QR 2 ω /4 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257298&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257298&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257299&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257299&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257299&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257300&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257300&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257300&type=middle 4 长方体(体分布) Q ( a 2 + b 2 ) ω /24 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257302&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257302&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257305&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257305&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257308&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257308&type=middle 5 实心圆柱体(体分布) QR 2 ω /4 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257310&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257310&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257312&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257312&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257314&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257314&type=middle 6 中空圆柱面 (面分布轴对称) QR 2 ω /2 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257316&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257316&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257319&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257319&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257319&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257322&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257322&type=middle 7 实心球 (体分布球对称) QR 2 ω /5 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257323&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257323&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257324&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257324&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257324&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257326&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257326&type=middle 8 薄空心球壳 (面分布球对称) QR 2 ω /3 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257329&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257329&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257330&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257330&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257331&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257331&type=middle 根据(8)和(12)式，矢量定轴磁矩 P m 及其时间的二阶导数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257332&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257332&type=middle m 正比于其标量定轴电矩 T l 。由 表 1 表 1 并结合(12)、(13)和(22)式可知，带电体在变速转动过程中，其远场辐射功率正比于定轴磁矩的二阶时间导数的平方，而其受到的阻尼力矩，正比于标量定轴电矩 T l 的平方，即与电荷相对于定轴的远近距离有关。具体而言，旋转带电体的阻尼力矩及其远场辐射功率，与带电体的特征尺寸( l 或 R 或 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257333&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257333&type=middle )的四次方成正比，在均匀分布情形，还与总电量 Q 的平方成正比。此外，辐射功率还与转动角速率的二阶时间导数的平方 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257334&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257334&type=middle 成正比；阻尼力矩还与转动角速的四阶导数 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257338&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257338&type=middle 成正比。另一方面，第3个示例中的均匀带电实心圆形平面和第5个示例中的均匀带电实心圆柱体，这两种情形下推导得到的公式相同，是因为带电体的电量与实心圆柱体的厚度成正比，换言之，圆柱体的厚度已经体现在总电量 Q 的计算中。 结论 1 由于定轴转动的带电体会产生磁矩，变速转动的带电体会产生变化的磁矩, 因此，转动体系会发生磁偶极辐射, 向外辐射能量。如无能量补充，由于带电体在变速转动过程中受到阻尼力矩的作用，它最终会趋于静止。首先, 本文在已有理论的基础上，推导了定轴变速旋转带电体的磁偶极辐射功率。然后，借助于带电体经历一个假定的任意周期性定轴转动过程，依据磁偶极辐射过程中能量守恒的原则，推导得到其磁偶极辐射阻尼力矩公式((22)式)。结果表明, 只有当角速度具有非零四阶导数( http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257338&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4257338&type=middle ≠0)时, 才存在辐射阻尼力矩。这一磁偶极辐射阻尼力矩又反过来影响旋转带电体的转动状态，使其转动振幅缓慢衰减，并使其转动圆频率略有减小。最后，本文给出了一些特殊形状变速旋转带电体的磁矩、磁偶极辐射功率与辐射阻尼力矩。 参考文献 虞国寅, 周国全.电动力学[M].武汉:武汉大学出版社, 2008: 87-88. 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Beijing: Higher Education Press, 2008: 200-205, 260-262(Ch). 周国全.一个矢量定理及其在物理教学与研究中的应用[J].物理通报, 2015(6): 32-35. DOI: 10.3969/j.issn.0509-4038.2015.06.010. ZHOU G Q. A vector theorem and its application in physics teaching and research[J]. Physics Bulletin , 2015(6): 32-35. DOI:10.3969/j.issn.0509-4038.2015.06.010(Ch). 10.14188/j.1671-8836.2019.01.014 O572;O442 whdxxblxb-65-1-105 高等学校电动力学课程教学研究项目(JZW-16-DD-15)；中央高校教育教学改革专项项目 2018-01-31 2019-02-24 2020-11-30 版权所有© 2019 《武汉大学学报(理学版)》编辑部 Copyright © 2019 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2019 武汉大学学报(理学版) 1 65