# Resistance between two points in an infinite metal sphere/cube

Let's imagine that we have a tridimensional metal object of infinite size, and decide to calculate the resistance between two arbitrary points. How would we go about doing this?

I have thought of two possibilities for calculating this, the first one being an adaptation of the conductivity (S/m) using the Pythagorean theorem to find a distance and isolating the resistance from there, treating it like we would treat an electrolytic liquid.

The second possibility I have been thinking about would be a 3-dimentional adaptation of the infinite-resistors problem (which I have only seen in 2d).

Would one of these methods result in giving a decent approximation, or would another method give better results?

## 1 Answer

For a homogeneous material characterized by a resistivity $\rho$ (in $\Omega m$) the resistance between any two points of contact is unbounded. Such "infinite" resistance even applies if one point of contact is replaced by a spherical contact area centered around the point. Just check for yourself and calculate the resistance for this latter configuration by integrating $\rho /(4 \pi r^2)$ from zero to any finite radial distance.

Another way of recognizing this divergence is by dimensional analysis. To get from a resistivity $\rho$ measured in $\Omega m$ to a resistance $R$ measured in $\Omega$, one has to divide $\rho$ by a length scale. This length scale can not be the distance between the contacts, as this would lead to the unphysical behavior of the resistance between two points decreasing with increasing distance. It turns our that the relevant length scale is the linear size $r$ of the electrical contacts: $R \approx \rho / r$.

Physically what happens is that the electrical field strength diverges towards a current injection point. You have to assume finite contact areas to obtain a meaningful answer.

• This doesn't seem correct. Ohm meters work: the result doesn't diverge or go to 0 if you put the two probes of an ohm meter on two positions of a sheet conductor - you get a nonzero reading for the resistance that does not depend sensitively on the shape of the contact area. – FrankH Jan 6 '13 at 12:23
• In 2D the divergence is milder (logarithmic), so you won't see a strong dependence on the size of the contact area. – Johannes Jan 6 '13 at 12:33
• This is true if the medium conductivity is much worse than that of the electrodes, then you'll indeed get the resistivity scaling as $1/r$ where $r$ is the electrode size. But what if the medium is made from the same material as the electrodes? Then you cannot even define the size of the electrode, right? The resistance between two points in a conducting object can always be found as a solution of a well-posed mathematical problem. Just consider a 3D grid of resistors (it is a possible 3D object), clearly there is finite resistance between any two nodes of the grid, no divergence of anything. – Maxim Umansky Aug 7 '13 at 4:50
• @MaximUmansky - 1) the electrodes need to have a much higher conductivity than the medium you are measuring, for the measurement to be interpretable. 2) the question is about a metal object, not a resistors network. But even for a resistors network, in the limit of smaller and smaller resistor sizes, you will observe the same divergence. – Johannes Aug 7 '13 at 14:59
• @Johannes - Yes, as I think about this more carefully I agree with you. Injecting a finite current through an infinitesimal area on the surface will lead to an infinite potential peak there; so the effective resistance becomes infinite indeed. I got confused with this because I was thinking in terms of a finite-difference solution to the Laplace equation, so a continuum body becomes sort of a network of resistors. But the numerical solution will not converge on a finer grid if the current injection area is not finite. Thanks. – Maxim Umansky Aug 9 '13 at 5:45