等势条件下两带电导体球的电荷分布

Charge distribution of two charged conductor spheres at equal potential

本文通过多重镜像法求解了等势条件下两个带电导体球的电荷分布问题,主要关注两球的电荷量以及平均面电荷密度随两球半径和两球间距的关系.通过选择合适的坐标,给出两导体球接触时,n级镜像电荷电量和位置的通式,及总电量的解析表达式.研究发现,当两导体球直接接触时,两球所带的电荷量可以严格求解,并给出了两球的电荷量之比表达式.随着两等势导体球间距的增大,两球所带的电荷量之比趋于半径之比.本文还讨论了一个导体球的半径趋于0的极限情况,小球与大球的电荷量之比趋于0,平均面电荷密度之比在两球不直接接触时趋于无穷大,而在两球直接接触时趋于π^(2)/6.

In this paper,the charge distribution of two charged conductor spheres at equal potential is solved by multiple image method.We mainly focus on the charge quantity distribution and the average surface charge density distribution as function of the radius and spacing of the two spheres.By choosing suitable coordinates,this paper gives the general formula of n-order mirror charge’s charge quantity and position,and the analytical expression of total charge quantity,when two conductor spheres are in contact.It is found that when two conductor spheres are in direct contact,the charge of the two spheres can be solved strictly,and the expression of the ratio of charge of the two spheres is given.As the distance between two equally potential conductor spheres increases,the ratio of the charges tends to the ratio of the radius.Considering the limiting case where the radius of one of the conductor spheres tends to 0,we find that the charge ratio of the small sphere to the large sphere tends to 0,and the ratio of the average surface charge density tends to infinity when the two spheres are not in direct contact,but tends toπ^(2)/6 when the two spheres are in direct contact.