Electrical Resistance of a Sphere?

You need to truncate the sphere by lopping off a small section at each end. That way the electrodes have some area. If a flat planar electrode pair is pressed against opposite sides of a true sphere, the resistance is infinite since the surface area between the sphere contact & electrode is literally zero. If the sphere ends are flattened, a finite area exists, & the resistance is finite.

Integration will compute the actual resistance. If a sphere is 10 cm in diameter, & the opposite ends of the sphere are lopped off resulting in 1.0 cm diameter contact surface area, you can compute resistance from those values. Did I help?

Claude

 

Claude, that is an approximation. Ut not exact because the material you excluded is in parallel and will result in lowering your R value. Conformal mapping might work for planar case (R of disk) followed by pi rotation. Numerical simulation will definitely work.

 

This implies that the potential present at the electrode is also present at the top of each cylinder, and likewise for the bottom--but that would only be true if the surface of each hemisphere were metallized. This problem is more complicated. You need to solve for the equipotential surfaces and streamlines throughout the solid, subject to the nonconductive boundary condition at the spherical surface.

 

No, it's quite consistent with what has already been pointed out. Your top cylinder grows narrower tending to 0 area as q tends to ∞. To get a finite resistance you will need to specify a minimum area of contact. Try q cylinders of height (r₀-d)/q.

 

Sorry, I misunderstood your cylinders to be concentric instead of disks.

 

Here it is computed, & the answer checks out. Not too hard. If you want to make it hard, offset the terminals at an oblique angle. That is tough. But this sheet computed resistance w/ the terminals diametrically opposite. That is too easy. A single integral solves it. Let me know if clarification is needed. Best regards.

Claude

 

No, I did not assume that. By symmetry the theta (angle in cylindrical system) vanishes due to z-axis symmetry. The radius "r', does appear in the integral. Take the sphere & divide into flat disk segments centered on the z-axis. The resistance of each disk is ∏c², where "c" is the radius of the flat disk. I computed said radius on the sheet.

Since the disks are stacked in series, we must sum all disks along the z-axis. The integral is a single integral wrt "z", but the radis of wach disk & that of the sphere enter into the computation. The polar angle theta does not enter in since we have symmetry. If the sphere had a section removed in a way that the angle was less than 2∏ radians, we would have theta in the integral.

Also, if we examine a section where b is half of a, we can compute the resistance of either cylinder with ease. I did that & demonstrated how the sphere section resistance falls in between the inscribed & circumscribed cylinders. That provides a sanity check but does not assure correctness. The resistances of the 2 cylinders are ρ/a multiplied by 0.318 and 0.424. The sphere section resistance must be in between these values. Although 0.350 lies in between 0.318 & 0.424, that itself is no guarantee. There are many values in between 0.318 & 0.424. However if my final answer was outside these values it would be wrong beyond a doubt. Since my answer lies in between these values, it might be correct.

Either way, it cannot be far off. We know the right answer is between the limits I just described. If I erred, please point it out. Did I miss a sign, drop a factor, make an invalid assumption, etc.? Just let me know & I will correct it. BR.

Claude

 

I have to agree with mfb. The problem with summing the resistances of flat disks is that you are neglecting the lateral spread of current from a small disk at top to a larger disk beneath it. Equipotential surfaces are not normal to the z axis (assuming that electrodes are at z=±a) but rather curve up in the upper hemisphere or down in the lower so as to intersect the spherical surface at right angles. Proper solution of this problem requires solution of Laplace's equation in spherical coordinates, with two sets of BC's specifying a) the source and sink surfaces, and b) zero normal component of current at the spherical boundary. I expect solutions for equipotential surfaces and for streamlines to be expressed in terms of Legendre polynomials.

I don't think this is such a simple problem.

 

But my math does account for lateral spread of current from smaller to larger disk. I do not believe you read my sheet carefully, but rather skimmed through it briefly. At the top plane, z=b, & there is a surface area, minimum at this plane & its corresponding plane diametrically opposite at z=-b. The area of the disk directly below the top disk is slightly larger than that of the top disk, resulting in lower resistance. The disks are stacked in series so that their resistance values add.

As the cross section changes, the lateral spread of current will encounter a changing area resuslting in changing resistance. I've covered that issue well. Now for the equipotential surfaces not being normal to z-axis, I say they are. Draw a diagram please & attach it for us to review. If you don't, I'll draw one in the evening. Equipotential surfaces intersect the z-axis normally due to symmetry. If, however, the terminal planes were not diametrically opposite, but offset/oblique, you would be correct. That is indeed a much more complex problem.

Please pay attention to my inscribed & circumscribed cylinder resistance values. They illustrate that I did indeed take the lateral current spread into account. Again, I am happy to accept correction, but please review my paper thoroughly. Skimming through it & making off the cuff rebuttals that are meritless gets us nowhere. I'm not out to "win" anything, an argument or otherwise, but I do notice that when someone, not necessarily me, produces info to a tough problem, there are always those who rebuke them while offering no evidence at all where the mistake is. I'm not accusing you of anything, just asking you to support your criticism. BR.

Claude

 

Claude, your equipotential surfaces are normal to the z axis because you forced them to be so by your choice of solution method. The resistance of a disk, likewise, is only [tex]R=\frac{\rho l}{A}[/tex] if the top and bottom surfaces are equipotentials, as though each were coated with a perfect metallic conductor--consistent with equipotentials normal to z. You can see easily that this cannot be a valid solution for the sphere. The BC (that current cannot flow across the sphere surface), together with Ohm's law [tex]\vec{J}=\sigma\vec{E}[/tex] means that [tex]E_{\perp}(r=a)=\left. \frac{\partial\phi}{\partial r}\right|_{r=a}=0.[/tex] The equipotential surfaces *must* be normal to the spherical surface, not flat as you postulate. It is the curvature of these surfaces that permits current to spread.

 

http://www.academia.edu/1841457/The_Notion_of_Electrical_Resistance_by_Soliverez

I will try to find time to check out his result with my simulator.

 

The lines should look similar to those for the TE11 mode in a spherical cavity resonator....