The principles of electrostatics are applied to dimensions both lower and higher than 3. Specifically, Laplace’s equations are solved in n dimensions subject to hyper-spherical symmetry in order to obtain the electric potential and hence the electric field. The physical problems associated with these solutions in 3-dimensional space are identified. The radial dependences of the potential and electric field are scrutinized. The successively lower radial dependences of the multipole fields are obtained by differentiating those of the multi-poles of the immediately lower order. The same results are also obtained by considering the hyper-surfaces of hyper-spheres in n dimensions. This study reaffirms the principles of electrostatics and provides a glimpse of the notion of higher dimensions.
Tan, Arjun and Edwards, Matthew E.
"Electrostatics and Dimensions of Space,"
Georgia Journal of Science, Vol. 68, No. 2, Article 8.
Available at: https://digitalcommons.gaacademy.org/gjs/vol68/iss2/8