# Capacitance with Earth?

• Narayanan KR
In summary, an insulated metal plate has been placed inside Earth (soil), and an voltage is applied between plate and Earth ground. The capacitance here will be modeled as a capacitor, but it will depend on the soil's relative permittivity and conductivity.f

#### Narayanan KR

1. In above image an insulated metal plate has been placed inside Earth (soil), and an voltage is applied between plate and Earth groung what will be the capacitance here??

2. one plate is the metal and the other plate is entire earth, so it is a big assymetrical Capacitor or What?

all replies welcomed friends.

I think it will fail to work as capacitor almost all supply voltage will occur across the insulation of the buried plate.

Narayanan KR
I think it can be modeled as a capacitor. The insulation will act like the dielectric between the plates...

Narayanan KR
The capacitance will be roughly twice that of a two plate capacitor eg..

C = 2*εRε0* A/d
where
ε0 is the permittivity of free space
εR is the relative permittivity of the insulation
A is the area of the plate
d is the thickness of the insulation.

You might need to include some "equivalent series resistance" due to the properties of the earth.

I do not think resistance of the Earth's crust will appear in the formula for capacitance. It will only affect the time for charging. The simplest equivalent circuit appears to be consisting of three plates separated by two insulating sheets: the outer plates will be connected to the negative terminal of battery and the central plate to the.positive terminal of the battery.

I think it will fail to work as capacitor almost all supply voltage will occur across the insulation of the buried plate.
if all votage appear across insulator, won't it try yo polarize it and store charge there by making capacitance?
...by the by i'll put another circuit in the thread with modifications

I do not think resistance of the Earth's crust will appear in the formula for capacitance.

It won't appear in the formula for capacitance, that's not what I meant. However you may need to include it in the model depending on what this is being used for.

The simplest equivalent circuit appears to be consisting of three plates separated by two insulating sheets..

See the drawing I posted. That's why there is a 2 in the equation.

Friends now apply my same question here, what will you think the capacitance will be ?.

( both the images are same except the distances between plates inside the ground )

I think it can be modeled as a capacitor. The insulation will act like the dielectric between the plates...

View attachment 99513
that's what I'm thinking Sir...please do see my second diagram also

Modelling wise you seem to be correct. But can we consider the whole soil of Earth as one conducting connection between two plates of a parallel capacitors.Just because we think the whole Earth is at same potential.

The capacitance will be roughly twice that of a two plate capacitor eg..

C = 2*εRε0* A/d
where
ε0 is the permittivity of free space
εR is the relative permittivity of the insulation
A is the area of the plate
d is the thickness of the insulation.

You might need to include some "equivalent series resistance" due to the properties of the earth.

Sounds to me like the right approach.
But why 2X ?
Seems to me we have something between a single plate capacitor and a two plate capacitor.
What's relative permittivity of topsoil where it's buried?
Moisture ought to affect that pretty strongly with its 80-ish dielectric constant
and conductivity will make it a lossy capacitor .

If the soil is highly conductive,
the model first posted is just like an electrolytic capacitor where you get a lot of capacitance
because the insulation is so thin -
it's just a few molecules of aluminum oxide on the surface of the aluminum plate.
Conductive electrolyte is the other "plate". Wet salty mud should behave similarly...
So i'll guess upper limit is half CWatters' answer.

If on the other hand the topsoil is not highly conductive, it becomes an insulating layer between the single plate capacitor and the rest of the universe . It's hard for me to imagine something with less permittivity than free space so i'll guess the lower limit is same capacitance as a single plate of same area surrounded by free space.

old jim

Narayanan KR
I agree with Hardy. As first step it is ok. Finally we will have to arrange in this manner and study what equivalent capacitance we get experimentally. Better to model after experimentation.

Because teh whole Earth can be considered to have the same potential hence that is equivalent to a single conductor, which is also equi-potential.

Because teh whole Earth can be considered to have the same potential hence that is equivalent to a single conductor, which is also equi-potential.

If your initial assumptions include that Earth is highly conductive medium , then yes it's equipotential.
Since that wasn't stated
and substantial potential difference can exist between points on Earth because it's only a mediocre conductor...

In your though experiment , tie down these loose ends before starting.

thank you mr. jim and let's think, i think i got some clearence

I actually got this idea on attempt to invent a overunity device.

" Assymetrical Capacitor is the Key to overunity"- Prof Donald Lee Smith.

I actually got this idea on attempt to invent a overunity device.

" Assymetrical Capacitor is the Key to overunity"- Prof Donald Lee Smith.

be careful where you take the thread - perpetual motion is a disallowed subject at PF.
If there's a quirk of math that leads one into over-unity engines - i'd say you should examine the derivation carefully for a divide by zero someplace.

https://en.wikipedia.org/wiki/The_Gods_Themselves

Have fun , and be rigorous with your thinking
"We think only through the medium of words. --Languages are true analytical methods. --Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method. --The art of reasoning is nothing more than a language well arranged. (Lavoisier)"
https://web.lemoyne.edu/giunta/ea/LAVPREFann.HTML
Well, i suppose one could argue we(at least some of us) think in pictures which we laboriously translate into words...
"But, after all, the sciences have made progress, because philosophers have applied themselves with more attention to observe, and have communicated to their language that precision and accuracy which they have employed in their observations: In correcting their language they reason better." ibid

old jim

Narayanan KR
be careful where you take the thread - perpetual motion is a disallowed subject at PF.
If there's a quirk of math that leads one into over-unity engines - i'd say you should examine the derivation carefully for a divide by zero someplace.

https://en.wikipedia.org/wiki/The_Gods_Themselves

Have fun , and be rigorous with your thinking

Well, i suppose one could argue we(at least some of us) think in pictures which we laboriously translate into words...

old jim
agreed Mr Jim. i won't do it again.

thank you mr. jim and let's think, i think i got some clearence

I actually got this idea on attempt to invent a overunity device.

" Assymetrical Capacitor is the Key to overunity"- Prof Donald Lee Smith.
Yes, as Jim said, discussions about PMMs and over-unity devices are not allowed at the PF. From the Forbidden Topics portion of the PF Rules (see INFO at the top of the page):
Pseudoscience, such as (but not limited to):
Perpetual motion and "free energy" discussions
http://wiki.4hv.org/index.php/Free_Energy_Debunking
http://en.wikipedia.org/wiki/Perpetual_motion
http://www.skepdic.com/freeenergy.html
http://www.skepdic.com/perpetual.html
The discussion about your buried capacitor and the conductance of soil is fine.

jim hardy
agreed Mr Jim. i won't do it again.
Thank you

Sounds to me like the right approach.
But why 2X ?
Seems to me we have something between a single plate capacitor and a two plate capacitor.

I treated it like a three plate capacitor with the two outer plates connected together. That's equivalent to two capacitors in parallel hence x2.

jim hardy
that's what I'm thinking Sir...please do see my second diagram also

Looks like you have an AC voltage source driving two capacitors in series.

Narayanan KR and Baluncore
Thanks CW...

now that i see your picture it's crystal clear...

Is this an AC or a low frequency DC circuit? The connection wires have significant capacitance due to their field. If it is a significant frequency AC, then it becomes a transmission line problem.

The good or bad conductivity of the soil is not really an issue as it is an equipotential. Resistance is in quadrature with capacitive reactance so it does not effect capacitance.

Re: top diagram of post #8. The two buried plates are in series, but the two sides of each are in parallel. That cancels to C = one surface plus wires.

The good or bad conductivity of the soil is not really an issue as it is an equipotential.

I'm not sure i can buy that.
It's only equipotential if it conducts, else there's an electric field across it which isn't possible if it's a conductor.

Let me change the shape from a flat plate to a sphere.
I need to do that because the capacitance of a single sphere is straightforward, see
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capsph.html
but the capacitance of a single plate is fearsome ,
http://arxiv.org/pdf/1304.6624v1.pdf
Abstract
The capacitance of arbitrarily shaped objects is reformulated in terms of the Neumann-Poincar ́e operator. Capacitance is simply the dielectric permittivity of the surrounding medium multiplied by the area of the object and divided by the squared norm of the Neumann-Poincar ́e eigenfunction that corresponds to its largest eigenvalue. The norm of this eigenfunction varies slowly with shape changes and allows perturbative calculations.

and i know my limits way too well to tangle with Poincaire operators and eignenfunctions .

So here goes:

If the soil is nonconductive and has permittivity not much different from that of free space
then an electric field exists between the surface of the sphere and the 'ground' wire that's also buried in the soil
and that volume holds whatever energy is stored in the capacitor.
If the distance to that ground wire is infinite then it's an isolated sphere with capacitance 4πε0R , per that hyperphysics link

If the soil is conductive, then
free charges in it migrate to the surface of the insulator and the electric field is constrained to the volume of the insulator
and capacitance will be εA/d = ε4πR2/d

so I hypothesize that conductive soil probably increases capacitance by ratio (ε4πR2/d ) / ( 4πεR ) = R/d

For OP's flat plate offering the difference between conducting and nonconducting soil might simplify to some reasonable function resembling √area / d
and that's why electrolytic capacitors work so well with one plate of metal, one plate of conductive semi-liguid paste, and insulation just a microscopically thin layer of oxide on the metal plate's surface. Gives a very small denominator.

That's what i think
and i'll welcome corrections. (See recent General Discussion thread "Have you ever failed"..)

old jim

1. In above image an insulated metal plate has been placed inside Earth (soil), and an voltage is applied between plate and Earth groung what will be the capacitance here??
I'm not sure that this shows the complete picture. Assuming attention has been given to ensuring soil conductivity, then in addition to the capacitance of your plate and cable combo, I think we'll also have a parallel capacitance of the entire Earth's sphere hanging off your buried cathode over on the right. If correct, this seems to place an extra 710uF in parallel to what you would calculate.

https://en.wikipedia.org/wiki/Capacitance

Narayanan KR
I'm not sure i can buy that.
It's only equipotential if it conducts, else there's an electric field across it which isn't possible if it's a conductor.
I notice it is now a DC system being modeled. The only high resistivity Earth's I have ever seen have been solid igneous rock. If the “soil” is an insulator then it is a most unusual soil as moisture and salt seem to be ever-present. Just the process of burying a sphere will create leakage paths.
, I think we'll also have a parallel capacitance of the entire Earth's sphere hanging off your buried cathode over on the right. If correct, this seems to place an extra 710uF in parallel to what you would calculate.
The electrostatic sphere of the Earth in space is not part of this circuit as the Earth is the reference.
The model can be simplified as two concentric spheres defined by, and separated by, the dielectric insulator.

NascentOxygen and jim hardy
If the “soil” is an insulator then it is a most unusual soil as moisture and salt seem to be ever-present. Just the process of burying a sphere will create leakage paths.

Quite so.

When posing a thought experiment i like to run it at its extremes then back off toward practicality .