Electrical Resistance of a Sphere?

[greswd]

How does one find the electrical resistance of a homogenous sphere of uniform density?

By connecting two wires across the diameter of the sphere, and assuming Pouillet's Law R=ρ\frac{L}{A} holds.

 

[mfb]

I don't think this is well-defined, if you try to calculate the resistance between two points on the sphere. If you have some finite area where you connect electrodes, you get some finite resistance value, but I think you will need a numerical simulation to calculate it.

 

[cabraham]

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

 

[marcusl]

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.

 

[greswd]

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.

But aren't the truncated parts in series, not parallel?

I tried numerical approximation but it leads to contradictory results. (See below)

 

[greswd]

First up, I have a solid sphere of radius r_0

I approximated by creating cylinders that "fill up" the sphere from inside, hence it is an under-approximation.


Let's look at one hemisphere. I split up the hemisphere into q cylinders of the same height.
q\inℤ^{+}


The height of each cylinder:
h=\frac{r_0}{q}


The radius of each cylinder, r_n
(r_n)^{2}+(r_x)^{2}=(r_0)^{2}
(r_n)^{2}+(nh)^{2}=(r_0)^{2}

(r_n)^{2}=(r_0)^{2}-(nh)^{2}
(r_n)^{2}=(r_0)^{2}-n^{2}\frac{(r_0)^{2}}{q^{2}}
(r_n)^{2}=(r_0)^{2}\left[\frac{q^{2}-n^{2}}{q^{2}}\right]


The area of each cylinder
A_n=π(r_0)^{2}\left[\frac{q^{2}-n^{2}}{q^{2}}\right]


Total Resistance (ignoring the last cylinder):

R_T=ρ\frac{h}{A_1}+ρ\frac{h}{A_2}+ρ\frac{h}{A_3}+...+ρ\frac{h}{A_{q-1}}
R_T=ρh×\sum_{n=1}^{q-1} \frac{1}{A_n}
\frac{1}{A_n}=\frac{1}{π(r_0)^{2}}\left[\frac{q^{2}}{q^{2}-n^{2}}\right]
R_T=\frac{ρh}{π(r_0)^{2}}×\sum_{n=1}^{q-1} \left[\frac{q^{2}}{q^{2}-n^{2}}\right]<br /> =\frac{ρ\frac{r_0}{q}}{π(r_0)^{2}}×\sum_{n=1}^{q-1} \left[\frac{q^{2}}{q^{2}-n^{2}}\right]<br /> =\frac{ρ}{πr_0}×\frac{1}{q}\sum_{n=1}^{q-1} \left[\frac{q^{2}}{q^{2}-n^{2}}\right]




R_T=\frac{ρ}{πr_0}×\frac{1}{q}\sum_{n=1}^{q-1} \left[\frac{q^{2}}{q^{2}-n^{2}}\right] , q→∞



That should be the resistance of a hemisphere. The problem is,

\frac{1}{q}\sum_{n=1}^{q-1} \left[\frac{q^{2}}{q^{2}-n^{2}}\right] , q→∞

does not converge to a single value. It apparently keeps on increasing as q inccreases. Hence, it would seem that a sphere can have any amount of resistance. This is quite contradictory.

 

[marcusl]

Total Resistance (ignoring the last cylinder):

R_T=ρ\frac{h}{A_1}+ρ\frac{h}{A_2}+ρ\frac{h}{A_3}+...+ρ\frac{h}{A_{q-1}}

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.

 

[haruspex]

This is quite contradictory.

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.

 

[greswd]

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.

Ah that explains it. Thanks haruspex.

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.

Why would there be no conductivity at the surface? Also, if I ignore the above, I should still be able to get an exact Pouillet value?

 

[marcusl]

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

 

[greswd]

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

that actually led me to come up with a new problem

 

[cabraham]

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

 

[mfb]

You seem to assume that voltage is a function of the z-coordinate only. Can you prove this assumption? I doubt that it is true.

 

[cabraham]

You seem to assume that voltage is a function of the z-coordinate only. Can you prove this assumption? I doubt that it is true.

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

 

[marcusl]

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.

 

[cabraham]

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

 

[marcusl]

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 R=\frac{\rho l}{A} 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 \vec{J}=\sigma\vec{E} means that E_{\perp}(r=a)=\left. \frac{\partial\phi}{\partial r}\right|_{r=a}=0. 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.

 

[the_emi_guy]

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.

 

[marcusl]

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

 

[mfb]

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

 

[marcusl]

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.

Thanks for finding this reference! I took a quick look and the approach looks right (note the Laplace equation in spherical coordinates in Eq. (29), whose solution is typically expressed in Legendre polynomials). His Fig. 3 shows believable streamlines and equipotentials, with EP's that are normal to the boundary as I said above. Another useful point is that the problem is simplified when the electrodes are taken to match equipotential surfaces, as at top and bottom of Fig. 3. As Soliverez points out, these approach hemispheres when they are small.

Here's the really interesting part: the resistance of the sphere ends up being just the sum of resistances of the top and bottom contacts. To see this, use the "spreading resistance" of a hemispherical contact pressed into a semi-infinite slab that was worked out ages ago (you see it in books on semiconductor contacts, for example), R_{contact}=\frac{1}{2\pi b \sigma} where b is the radius of the hemispherical contact. Double it to account for top and bottom contacts in series and you get Soliverez's Eq. (43)! We should have realized from the very start that, for b<<a, the spherical conductor can be considered infinite in comparison to the size of the contacts, so all significant resistance is contributed by the small contact areas.

 

[cabraham]

marcusl - I don't understand how you can dispute me when the link above to Carlos' paper gives the same answer I gave. In his math, he uses "z₀" to denote the terminal location on the z axis, where I use "b" for the same. However, I use "c" as the radius of the circular cross sectional area intercepted by the terminal plane, where Carlos uses "b". His "b" is not the same as my "b'. Carlos' "b" is Claude's "c", & Carlos' "z₀" is Claude's "b". Also his equation has a factor of 2 in the denominator where mine doesn't. Carlos' integral is over the whole sphere top to bottom, where Claude's (moi) is center to top with a factor of 2 multiplied in. That is the descrepency in the 2 equations. If you compute the R value using Carlos' equation & Claude's you should get the same answer.

Notice his natural log term. The argument has "a+z₀" in the numerator, with "a-z₀" in the denominator. Same as mine. Examine carefully & his answer is identical to mine. He uses conductivity "sigma", where I use resistivity "rho". Please review carefully. Carlos & I agree. BR.

Claude

 

[cabraham]

Also, the equation:

R = ρl/A

is valid over the region of integration. Laplace's equation is certainly upheld here, but it is a long roundabout way of computing the resistance. Carlos is indeed accurate, but my approach gets the same answer with much less effort.

However, for the situation I described at the end of my sheet, where the terminals are not diametrically opposite, but oblique, then Carlos' approach is exactly how I would tackle the problem. Again, from looking his equation over, we seem to agree.

marcusl - Would you please recompute the examples you gave above while duly noting that "b" in Carlos' method is not my "b" To convert, remember that a "b" of 0.1 in Carlos' math, with a radius "a" of 1 for the whole sphere, then we must use Pythagoreus' theorem. My "b" would be sqrt(1 - (0.1)²) = 0.995, quite a difference. As Carlos' "b" becomes vanishingly small, my "b" approaches unity (if sphere radius "a" is unity).

Please make due note of this important distinction. Thanks to all for your feedback.

Claude

 

[cabraham]

Thanks for finding this reference! I took a quick look and the approach looks right (note the Laplace equation in spherical coordinates in Eq. (29), whose solution is typically expressed in Legendre polynomials). His Fig. 3 shows believable streamlines and equipotentials, with EP's that are normal to the boundary as I said above. Another useful point is that the problem is simplified when the electrodes are taken to match equipotential surfaces, as at top and bottom of Fig. 3. As Soliverez points out, these approach hemispheres when they are small.

Here's the really interesting part: the resistance of the sphere ends up being just the sum of resistances of the top and bottom contacts. To see this, use the "spreading resistance" of a hemispherical contact pressed into a semi-infinite slab that was worked out ages ago (you see it in books on semiconductor contacts, for example), R_{contact}=\frac{1}{2\pi b \sigma} where b is the radius of the hemispherical contact. Double it to account for top and bottom contacts in series and you get Soliverez's Eq. (43)! We should have realized from the very start that, for b<<a, the spherical conductor can be considered infinite in comparison to the size of the contact so all significant resistance is contributed by the small contact areas.

My approach says the same. Just note the distinction that my "b' differs from that of Carlos. For a Carlos "b" of 0.10, the equivalent Claude "b" is 0.995, & the resistance is ρ/a times 1.9055, over 5 times greater than the 0.350 value when Carlos "b" is 0.8660 with Claude's "b" at 0.5 times a.

But if Carlos "b" decreases to 0.01 times "a", we get a Claude "b" of 0.99995. My resistance is then equal to ρ/a times 3.3730. As the radius of the termination decreases, the overall resistance increases due to smaller terminal contact area, just as the discussions above indicate. Anyway I just thought I would mention it. BR.

Claude

 

[marcusl]

Claude, we're still not seeing eye to eye.

Your expression for resistance in the attachment to post #12 is R=\frac{\rho}{\pi a} \log \frac{a+z_0}{a-z_0} where I've used z_0 in place of b just to be consistent with usage in my previous post. This can be rewritten as R=\frac{\rho}{\pi a} \log \frac{1+z_0/a}{1-z_0/a}=\frac{\rho}{\pi a} \log (1+z_0/a)- \log (1-z_0/a). Since z_0/a is less than one, we can apply the expansion \log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}... to get
R=\frac{2\rho}{\pi a} \left[\frac{z_0}{a} + \frac{1}{3}\left(\frac{z_0}{a}\right)^3 + ...\right]. Substituting z_0=\sqrt{a^2-b^2} for z0, where b is the contact radius, gives
R=\frac{2\rho}{\pi a} \left[ \sqrt{1 - \left(\frac{b}{a}\right) ^2} +<br /> \sqrt[3]{1 - \left(\frac{b}{a}\right) ^2 } + ...\right].
Comparison to Soliverez's expression
R=\frac{1}{\pi \sigma b} shows a poor match--the a's and b's aren't even in the same places.

Finally I invite you to work out a numerical example like rho=sigma=1, a=1, contact radius b=c=0.01 and z0=0.99995.
Soliveres gets R=31.8 ohms, your expression gives 3.4 ohms. mfb was pointing out a similar discrepancy with his lower bound calculation.

 

[cabraham]

Claude, we're still not seeing eye to eye.

Your expression for resistance in the attachment to post #12 is R=\frac{\rho}{\pi a} \log \frac{a+z_0}{a-z_0} where I've used z_0 in place of b just to be consistent with usage in my previous post. This can be rewritten as R=\frac{\rho}{\pi a} \log \frac{1+z_0/a}{1-z_0/a}=\frac{\rho}{\pi a} \log (1+z_0/a)- \log (1-z_0/a). Since z_0/a is less than one, we can apply the expansion \log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}... to get
R=\frac{2\rho}{\pi a} \left[\frac{z_0}{a} + \frac{1}{3}\left(\frac{z_0}{a}\right)^3 + ...\right]. substituting z_0=\sqrt{a^2-b^2} for z0, where b is the contact radius, gives
R=\frac{2\rho}{\pi a} \left[ \sqrt{1 - \left(\frac{b}{a}\right) ^2} +<br /> \sqrt[3]{1 - \left(\frac{b}{a}\right) ^2 } + ...\right].
Comparison to Soliverez's expression
R=\frac{1}{\pi \sigma b} shows a poor match--the a's and b's aren't even in the same places.

Finally I invite you to work out a numerical example like rho=sigma=1, a=1, contact radius b=c=0.01 and z0=0.99995.
Soliveres gets R=31.8 ohms, your expression gives 3.4 ohms.

marcusl - Please recheck your log \frac{1+z_0/a}{1-z_0/a}= ---

<<your quote: Since z_0/a is less than one, we can apply the expansion \log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}... to get --- end quote>>

The quantity "z₀/a is hardly less than 1. In fact it is almost equal to 1, for a contact radius of 0.1, z₀ equals 0.9995, hardly less than 1 at all. Expanding the log function into its Taylor series requires many terms before convergence.

Your 2nd mistake is near the end. After replacing z₀ with sqrt{a^2-b^2}, you get a quantity inside the brackets consisting of radicals containing 1 - (b/a)². But this quantity is virtually 1 for b very small. Hence the radicals all have approx. unity value. But you put coefficients of unity before each radical. The coefficients should have a progression of:

1, 1/3, 1/5, 1/7, ---

The sum of said coefficients diverges. You erred at the top when you took my log expression, then did the Taylor series. I will re-examine this, byt frankly, my inscribed/circumscribed cylinders seem to support my final expression. I will examine later. Right now my next higher household authority is on me to put up the Christmas tree. I will get back to you later tonight. BR.

Claude

 

[marcusl]

The typo in omitting the numerical factor 1/3 from the expression towards the end of my post doesn't change the conclusion. Try the numerical example I gave above, and see that results from the two methods differ by an order of magnitude.

If you use the spreading resistance for a disk instead of a hemisphere to better match your contact geometry
R\approxeq 2 R_{disk}=\frac{1}{2\sigma b} then the discrepancy is even larger.

 

[cabraham]

The typo in omitting the numerical factor 1/3 from the expression towards the end of my post doesn't change the conclusion. Try the numerical example I gave above, and see that results from the two methods differ by an order of magnitude.

If you use the spreading resistance for a disk instead of a hemisphere to better match your contact geometry
R\approxeq 2 R_{disk}=\frac{1}{2\sigma b} then the discrepancy is even larger.

Sorry marcusl but I'm convinced it changes the conclusion by a lot, because omitting 1/3, 1/5, etc. results in convergence as opposed to divergence. Anyway I composed 2 Excel spread sheets using the summation of stacked disks. As we subdivide into finer increments we get better accuracy. I used 99 disks per hemisphere, then 989. As expected the 99 piece answer was 8% off, & the 989 piece answer was 1% off from my equation. Unfortunately the larger sheet file is too large to attach but I attached the smaller file.

I do not believe that you can replace my log term with a very short truncated Taylor series because the z0 quantity (my "b") is nearly unity. Please repeat your computations w/o such replacement, i.e. let a log function remain as writted w/o trying to simplify. Carlos did very well but his assumptions that this term is negligible & can be dropped might not be valid. Please examine my spread sheet. Thanks for your interest.

Claude

 

[marcusl]

You are missing the points (many of them). Your exact expression (not my expansion) is an order of magnitude too small compared both to mfb's lower bound and to Soliverez's expression.

I am done with this topic.

 

[cabraham]

You are missing the points (many of them). Your exact expression (not my expansion) is an order of magnitude too small compared both to mfb's lower bound and to Soliverez's expression.

I am done with this topic.

Is it possible that Soliverez' equation is an order of magnitude too high? Without proof you accept his, not mine. The Excel spread sheet confirms my answer. By manual summing each disk section we get answers much closer to mine than to CS. I guess it's pointless to argue. Nobody has been able to show where I went wrong other than to say my result is "too large" as if CS result is somehow immutable.

Claude

 

[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 throught 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