# Modified Coulomb's law (hypothetical) and conductor

I came with problem where it was asked to comment on the charge distribution and electric field inside a solid conductor sphere, if the Coulomb's, instead of following inverse square relationship, follows an inverse cube relationship.

I thought that fact that will remain invariant irrespective of Coulomb's law is that there will not be any flow of charge within an isolated charged sphere , hence if we assume that there is some non zero charge density within the sphere then charges within the bulk of sphere, then charges at two different points in the bulk will would experience some force and would align themselves so as to attain the state of minimum energy,and that would preferably be when they are on the surface and can't move more farther apart. Also it can be argued that presence of charges that are in the bulk would lead flow of charge which will continue to flow until every point on the sphere is equipotential. Also since it is known that presence of electric field in a conductor, induces another electric field inside it which nullifies the net electric field will still be true as it due to the nature of conductor (presence of free charges), hence any electric field produced by the given charge will be nullified by e- of the conductor, hence field inside the conductor would remain zero (to prevent flow of charges and infinite heat dissipation).

The problem is that I don't really know if the conclusion is correct and would gladly love some input and ideas regarding my interpretation or some new insights into what I was missing or where I am flawed.My interpretation might seem naive please be kind enough to pardon.

Using successive (induced) image charges for two charged conducting spheres, some results can be obtained:

\begin{align} \begin{pmatrix} V_a \\ V_b \end{pmatrix} &= \tfrac{1}{4\pi \epsilon_0} \begin{pmatrix} \frac{1}{a}-\frac{b^3}{d^4}-\frac{b^5}{d^6}-\ldots & \frac{1}{d}+\frac{2a^3 b^3}{d^7}+\ldots \\ \frac{1}{d}+\frac{2a^3 b^3}{d^7}+\ldots & \frac{1}{b}-\frac{a^3}{d^4}-\frac{a^5}{d^6}-\ldots \end{pmatrix} \begin{pmatrix} Q_a \\ Q_b \end{pmatrix} \\[5pt] U &= \tfrac{1}{2} \begin{pmatrix} Q_a & Q_b \end{pmatrix} \begin{pmatrix} V_a \\ V_b \end{pmatrix} \\[5pt] &= \tfrac{1}{8\pi \epsilon_0} \left( \tfrac{Q_a^2}{a}+\tfrac{Q_b^2}{b}+\tfrac{2Q_a Q_b}{d}- \tfrac{a^3 Q_b^2+b^3 Q_a^2}{d^4}- \tfrac{a^5 Q_b^2+b^5 Q_a^2}{d^6}+\tfrac{4a^3 b^3Q_a Q_b}{d^7}-\ldots \right) \\[5pt] F &= \tfrac{1}{4\pi \epsilon_0} \left[ \tfrac{Q_a Q_b}{d^2}- \tfrac{2(a^3 Q_b^2+b^3 Q_a^2)}{d^5}- \tfrac{3(a^5 Q_b^2+b^5 Q_a^2)}{d^7}+ \tfrac{14a^3 b^3 Q_a Q_b}{d^8}-\ldots \right] \end{align}

As a result, it'll be more attractive for unlike charges ($Q_a Q_b<0$) and less repulsive for like charges ($Q_a Q_b>0$).

See more in this paper or this answer and also

• $\underline{\text{Problem }2.6}$ in Jackson JD. Classical electrodynamics. 3rd ed. New York: John Wiley and Sons; 1998.

• Soules JA. American Journal of Physics 58, 1195 (1990).

• $\underline{\text{pp. 128-131}}$, William R Symthe. Static and Dynamic Electricity. 3rd ed. New York: Hemisphere Publishing Corporation, 1989