Electrical Resistance of a Sphere?

[mfb]

Is it possible that Soliverez' equation is an order of magnitude too high? Without proof you accept his, not mine.

We have two independent derivations (at the time I wrote my post, the link to Soliverez' was not there yet), both agree with each other.

On the other hand, we have your calculation which is based on an incorrect assumption.

The Excel spread sheet confirms my answer.

It uses the same, wrong assumption as your analytic approach.

Nobody has been able to show where I went wrong

We did so several times, and we ran out of different ways to explain it to you. You just ignore those explanations and examples.

 

[cabraham]

We have two independent derivations (at the time I wrote my post, the link to Soliverez' was not there yet), both agree with each other.

On the other hand, we have your calculation which is based on an incorrect assumption.


It uses the same, wrong assumption as your analytic approach.


We did so several times, and we ran out of different ways to explain it to you. You just ignore those explanations and examples.

Just where did I go wrong? You haven't answered. I will review your posts carefully. What is wrong with summing the stacked disks in series. Each disk has a differing area so that R differs. Current laterally diffuses over differing areas, & ther disks are summed. It has to be right. Again, I will carefully review, but my 2 cylinder boundary limit affirms my answer. BR.

Claude

 

[cabraham]

You underestimate the total resistance with your approach as you do not consider current flow in radial direction (which contributes to resistance, too). This is easier to see if you take extreme cases - attach two small disks of radius r (the electrodes) to a cylinder with radius R>>r and height h~r. If you increase R, your calculated resistance would drop with 1/R^2, but the real resistance will approach a finite value.

I did some two-dimensional numerical simulation, but the borders are messy to consider.
For b=0.96, I get ~2.3 ρ/a as resistance, where your formula gives 1.24. I think my simulation overestimates the resistance a bit, so the real value is somewhere in between.
For b=0.52, I get ~0.38 ρ/a as resistance, where your formula gives 0.367. Here, the radial resistance is not significant.

Fixing a=ρ=1 now, as they are just prefactors anyway:

For (b-1)<<1, it is possible to calculate an interesting lower bound of the resistance as following:
Replace the disk (electrode) by a half-sphere. This replaces conducting material by an ideal conductor and clearly reduces the resistance.
Replace the half-ball by a half-ball around the electrode and connect the second electrode to the outer boundary. Use this as half the full resistance. This adds conducting materials where the original problem has none and moves the original symmetry area towards the electrode, so it again underestimates the resistance.
This modified problem has a spherical symmetry, so we can solve it with a one-dimensional integral. Let r be the radius of the electrode, $r^2+b^2=1$.

$$R > 2 \int_r^1 \frac{1}{2\pi r'^2} dr' = \frac{1}{\pi} (\frac{1}{r}-1) = \frac{1}{\pi}(\frac{1}{\sqrt{1-b^2}}-1)$$

Compare this with your formula at b=0.99:
R > 1.93
R with your formula: 1.68

It gets worse with larger b, as the radial flow gets more important:
b=0.9999:
R > 22.2
R with your formula: 3.15

I think I see the discrepency. Your "radial flow" would seem to originate from a point source contact terminal. I assumed, maybe incorrectly, that the contacts were flat planes w/ circular cross section, & that the electrodes from the power supply contacted the entire area of the sphere terminations. Here the current is primarily "vertical" w/ lateral diffusion (spreading) as charges transit downward.

But you mention "radial flow", & I am thinking that at the contact area, the current would not be normal to the equipotential contact surface. Right at the contact surfave the contact area is in fact an equipotential plane. Hence all current entering/exiting the sphere section must be normal to that contact plane as it is an equipotential surface. If there was a component of current oriented radially, then we would have current at an oblique angle to an equipotential surface, which I didn't think can happen.

Can you provide a quick hand sketch of current paths & equipotential surfaces detailing what you describe as "radial flow". Thanks.

Claude

 

[mfb]

What is wrong with summing the stacked disks in series.

It requires perfect radial flow of current.
It gives wrong predictions for the surface of the ball (equipotential areas have to be perpendicular to the surface, as shown above - they are not in your model).
It gives wrong predictions for the total resistance (see my lower bound derived above).
It gives wrong predictions for the limit of disks with radius R->infinity.

Again, I will carefully review, but my 2 cylinder boundary limit affirms my answer.

No, it just shows that your answer is between some lower and upper bounds - which are very weak for small contact areas.

Your "radial flow" would seem to originate from a point source contact terminal.

No, point sources are unphysical. I used small hemispheres.

If there was a component of current oriented radially, then we would have current at an oblique angle to an equipotential surface, which I didn't think can happen.

You need a radial component "somewhere" to have current flow everywhere in the sphere. In fact, you get radial flow everywhere, with the most significant part close to the edge of the electrodes.I attached a sketch of a sphere, the vertical lines are current, the horizontal lines are equipotential surfaces. You can see how they are bent. The right/upper part is symmetric to the lower left part. The uppermost and lowermost horizontal line are the electrodes.
- equipotential surfaces have to be perpendicular to the surface of the ball
- current has to be perpendicular to the equipotential surfaces.

 

[Studiot]

I have always understood the analytical solution (current isolines) is the same as the solution for foundation pressure trajectories in the ground or stress trajectories from a point load in elasticity, first presented by Coulomb.

 

[cabraham]

I am re-examining my stacked disk model, & it looks like I will have to deep six it. I presumed at first that current would laterally diffuse & distribute evenly through each disk cross section. That is approximately true if we are near the center of the sphere & equipotential surfaces are almost flat. But near the poles, the curve is too pronounced for even distribution. I think the "UD" model (uniform density) results in too low an R value.

I will try to solve the Laplace equation as soon as I figure out how. It will require general curvilinear coordinates. I will post when I have the solution. BR.

Claude

 

[Redbelly98]

Not sure if this would help in the modeling, but I was thinking along the lines of assuming small spherical electrodes, each centered on opposite sides of the conducting sphere. The E-field must be parallel to the edge of the conducting sphere everywhere***, so there would be a surface charge on the sphere's surface to force that to happen.

I'm hard-pressed to spend time on solving this, but if I did I'd be more inclined to go the numerical approximation route rather than grinding it out algebraically.

*** If the E-field is not parallel to the surface, then it is either pushing charge into the surface or removing charge from the surface, thereby changing the surface charge density. In steady state the surface charge doesn't change in time, so E must be parallel to the conductor's surface everywhere, in steady state.

 

[CarlosSoli]

I wish to stress that my work, "The notion of electrical resistance" is a general discussion of the concept, where the calculation of the resistance of a sphere is only an illustration. I will gladly discuss here or elsewhere any details of it, though I point out that I have not followed the details of the previous discussions. If anyone finds any error in my calculations, I will be glad to correct them and acknowledge the contribution in the following editions of the document, which is published under a Noncommercial-Share Alike 3.0 Unported license (free use, modification and distribution, acknowledging original author).

Carlos Solivérez

 

[Menaus]

If you guys want even more mathematical fun, you can try calculating the resistance if this sphere is under resonance, let's make it pulsed DC to make things simpler. :p