21. 电和磁有许多相似之处,比如同种电荷相斥,同名磁极也相斥,异种电荷相吸,异名磁极也相吸:变化的电场激发磁场变化的磁场也能激发电场。著名的英国物理学家狄拉克曾预言磁单极子可以像正负电荷一样独立存在,所谓磁单极子是指仅带有N极或S极单一磁极的磁性物质,它们的磁感线分布类似于点电荷的电场线分布(如图甲)。若存在一N极磁单极子如图乙所示,固定于水平桌面上,其上覆盖一半径为r的半球壳,球心恰好在磁单极子处,球壳表面绝缘光滑。现在半球壳上方轻轻放置一特殊材料制成的质量为m的弹性导电线圈,线圈开始时半径为
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3Er%3C%2Fmi%3E%3Cmn%3E0%3C%2Fmn%3E%3C%2Fmsub%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmi%3Er%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,电阻为R,线圈在重力作用下沿着半球壳下滑,下滑过程中线圈平面始终在水平面内,当线圈下滑高度h=0.2r时速度为v,已知磁单极子距离为d处的磁场强度大小为
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3EB%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmi%3Ed%3C%2Fmi%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,弹性线圈存储的势能与橡皮筋类似,当线圈长度伸长∆x时存储的弹性势能为
![](http://math.21cnjy.com/mml2svg?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmsub%3E%3Cmi%3EE%3C%2Fmi%3E%3Cmi+mathvariant%3D%22normal%22%3Ep%3C%2Fmi%3E%3C%2Fmsub%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmfrac%3E%3Cmsup%3E%3Cmi%3Ek%3C%2Fmi%3E%3Cmn%3E%27%3C%2Fmn%3E%3C%2Fmsup%3E%3Cmi%3E%CE%94%3C%2Fmi%3E%3Cmsup%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmn%3E2%3C%2Fmn%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,上述式中k、k′均为常数,求: