Question

Derive coulomb's law in five dimensions.

Answer 1

Fifth Dimension

From fifth dimension, we begin to see the higher dimensions and it deals with the subatomic level. These dimensions are curled in on themselves and it deals with several possibilities. In particular, in dd dimensions, we have that

over the boundary of some spatial region only if the field of a point charge has the form

where is measured as charge per unit n-volume. r is the distance between the charge Q₁ and Q₂. I believe there are no stable bound atoms in dimensions n>3 with the usual electromagnetic laws. The power dependence of r^-3 for four dimension space leads the Schrodinger equation having energy levels unbounded. It will not save an atom from collapse.

All the things presented above are the simplest consideration. If you consider more detailed theory you can find that Space–Time–Matter theory can be possible. In this theory, the fifth dimension is not temporal or spatial but mass-like, i.e. position in the fifth dimension is related to the mass of the particle. Standard Model gauge fields and fermions could also propagate in the five-dimensional space–time. Some researcher showed that the electric charge as the fourth component of the particle momentum, i.e. the fifth component of the five-vector energy-momentum. As a consequence, Coulomb’s electrostatic force only emerges from the 5D gravitational force. We can obtain Coulomb’s electrostatic force in addition to Newton’s gravitational force and it would look like a “classical version” of Kaluza–Klein results. The net force between the charge will be in fifth dimension

where m₁ and m₂ are the masses of the two charges, u is unit vector in the direction of their positions.