Definition of ground potential in infinite sheet charge distributions

[amylostorm]

A few rules are commonly taught when dealing with problems regarding charge distributions of multiple charged plates/sheets at specified distances from each other(the sheets have infinite size) -

• Any two sheets facing each other must have opposite charges equal in magnitude on the inner faces
• The outer-most faces of the outer-most sheets must have charge equal to half the sum of all the charges distributed on all plates
• When any one sheet is "grounded", the sum of the charges of all plates becomes zero, thus the charge on the outer-most faces of the outer-most sheets is also zero

These rules can be easily derived from the fact that conductors cannot have electric fields inside them as shown here - electrostatics - Why are the two outer charge densities on a system of parallel charged plates identical? - Physics Stack Exchange & electrostatics - Grounding system of conducting plates - Physics Stack Exchange

Now for my question, what exactly would grounding mean in such a system? Is it at 0V with respect to something? Or did we assume it as our reference potential? If we did the latter then I think the solution to these problems would also depend on its "absolute potential" as in say we have n plates with charges $Q_1 , Q_2 ,..., Q_n$ and we connect one of these to a conducting hollow sphere far enough that it is not affected by the fields of this system. This is our ground. Then depending upon its potential($\frac{kq_{sphere}}{R}$) wouldn't the sum of the charges on all the plates differ from zero? If it had a very high potential then it would force positive charge into the plate even if $\sum Q_i$ was already positive. This could be done even while respecting the $E=0$ condition of all the plates. eg -

By varying x in these any amount of positive charge could be delivered into B while also having the fields cancel out inside of all conductors.

Tl;DR - explain a ground's potential and function in the context of infinite charged plates and prove that the sum of charges of all plates must be zero if one of them is grounded.

 

[TSny]

Now for my question, what exactly would grounding mean in such a system? Is it at 0V with respect to something? Or did we assume it as our reference potential? If we did the latter then I think the solution to these problems would also depend on its "absolute potential" as in say we have n plates with charges $Q_1 , Q_2 ,..., Q_n$ and we connect one of these to a conducting hollow sphere far enough that it is not affected by the fields of this system. This is our ground.

When none of the plates is grounded, there will be charge on the outer surfaces of the outer plates, in general. So, there will be a uniform electric field that extends to infinity from the outer surfaces of the outer plates. It will be impossible to introduce a conducting hollow sphere far enough away to be unaffected by these fields. Also, there will not be a definite value of potential "at infinity". So, defining "grounding of a plate" as setting the potential of the plate to some value is not going to be meaningful (I believe).

For a system of finite-sized conductors, we can take the potential "at infinity" to be zero and define grounding a conductor as setting the potential of the conductor to zero. I believe you can show that the final charge on the grounded conductor is such as to minimize the total electrostatic energy of the system with respect to varying the charge on the grounded conductor.

For the system of infinite plates, I would interpret "grounding a plate" to mean that you allow the charge on that plate to assume the value that minimizes the total electrostatic potential energy of the system. This was mentioned in your links to Physics Stack Exchange.

 

[amylostorm]

For a system of finite-sized conductors, we can take the potential "at infinity" to be zero and define grounding a conductor as setting the potential of the conductor to zero. I believe you can show that the final charge on the grounded conductor is such as to minimize the total electrostatic energy of the system with respect to varying the charge on the grounded conductor.

That is something I did not understand while reading that answer on stackexchange, how does making the potential of a plate zero(w.r.t. infinity) correlate with its energy getting minimized? I don't even know if the problem is answerable as it would be hard to calculate the potentials of finite plates w.r.t. infinity(let alone set one to zero) due to the fringing fields as they are finite sheets. Plus, wouldn't the solution of minimizing energy only work for grounds with a potential w.r.t. infinity of 0? What if we connected it to my hollow charged sphere?

As a sidenote, I find it weird that in EE the ground is taken(by setting it as reference) as 0V and all measurements are done w.r.t. to it but in physics we still take infinity to be zero and force the ground to be at 0 V w.r.t infinity. Can you shed some light on why this difference exists?

 

[alan123hk]

As a sidenote, I find it weird that in EE the ground is taken(by setting it as reference) as 0V and all measurements are done w.r.t. to it but in physics we still take infinity to be zero and force the ground to be at 0 V w.r.t infinity. Can you shed some light on why this difference exists?

I'm not very familiar with the the difference in customary interpretations of ground between physics and electrical engineering.

But regardless, this is just a common reference point for us to communicate and deal with problems. If we don't use a common reference point, then every communication has to explain which two points a potential difference is relative to, which is really annoying. Wouldn't it be simpler and more consistent with most practical applications if we had a common reference point (zero volt) and we could directly say how much potential another point has?

In a system, multiple points or locations can be defined as ground, meaning they are all zero volts. But of course, we have to make sure that in practical applications they can indeed be considered without potential differences.

https://www.physicsclassroom.com/ge...ding/Lecture-Notes/LessonNotes.pdf?lang=en-US

 

[amylostorm]

@alan123hk I mean yeah, I did word that in a very weird way. You are of course correct and what I meant was that in EE we can connect a circuit to a ground at $V$ potential(w.r.t. to some reference) and it's behaviour doesn't change. While in this problem changing the ground potential should(according to me) change the distribution of charge in the system.

Also, is it necessary to take ground as a potential reference every single time? The way I see it, grounding a body just means connecting it to a large conducting body kept far away, like the earth. That is a physical process, while assuming that body's potential as 0 volts is just semantics. Am I wrong?

 

[Steve4Physics]

Also, is it necessary to take ground as a potential reference every single time? The way I see it, grounding a body just means connecting it to a large conducting body kept far away, like the earth. That is a physical process, while assuming that body's potential as 0 volts is just semantics. Am I wrong?

Here are some comments about potentially (ha ha) ambiguous terminology...

1. In some contexts, ‘ground’ (or ‘earth’ or ‘chassis’) is simply used to mean a convenient fixed voltage reference level, taken as 0V, in a circuit. E.g. the negative terminal of a car’s electrical system.

2. In some contexts, 0V is taken as the potential at infinity (where the electric field from all charges is zero). But this is not then referred to as ‘ground’.

3. In some contexts, ‘ground’ refers to the earth’s surface – treated as a conductor at 0V; connections to ‘ground’ are used to facilitate electrical safety.

4. In some contexts, where current is required to flow between some apparatus and ‘ground’ (such as the plate being "grounded" in Post #1), there is an implicit requirement that ‘ground’ must remain at a constant potential (which is taken as 0V for convenience). Ground can then be conceptualised as a plate of an ideal infinite capacitor – charge can enter or leave it without changing its potential (as you suggested in Post #5).

(Note that the earth itself, if treated as a conducting sphere, has a capacitance of about $710 \mu F$ which is surprisingly small.)

The above is not meant to be rigorous/complete – it's just for clarification.

Minor edit.

 

[alan123hk]

EE we can connect a circuit to a ground at V potential(w.r.t. to some reference) and it's behaviour doesn't change. While in this problem changing the ground potential should(according to me) change the distribution of charge in the system

There may be some communication issues here. For example, for a given circuit diagram, the values of all resistors, capacitors, resistors and their connection methods are determined and cannot be changed, so of course you can choose any point as a ground when analyzing the circuit diagram. But I believe EE won't say that you can ground any point in any real circuit without changing the distribution of charge throughout the system, which is obviously impossible because the charge in the grounded conductor will be discharged or neutralized.

Also, is it necessary to take ground as a potential reference every single time?

Of course, this is not necessarily the case. For example, those electric mosquito swatters only indicate how many volts the high voltage is, but do not define where the ground is and how many volts the voltage is relative to the ground.

The way I see it, grounding a body just means connecting it to a large conducting body kept far away, like the earth. That is a physical process, while assuming that body's potential as 0 volts is just semantics. Am I wrong?

I don't think it necessarily needs to be very far. We can take one conductor in the system as a ground reference, and all absolute voltages are measured relative to that conductor. Otherwise, we could also connect the conductor to a larger object, one with almost infinite capacitance. Then charge can freely removed or added to ground without changing its potential. It is tradition and consensus to set the ground potential to 0, which is also the most convenient option.

 

[alan123hk]

Some further personal thoughts on "Ground".
The definition of ground is generally considered to be that it can absorb or supply current without changing its potential, in other words, ground should be a perfect sink or source for current, so it is also an equipotential point or plane which serves as a reference for the circuit considered. Although this definition of ground, like ideal voltage and current sources, is in reality an unrealistic fiction, we hope that it can still serve as a guiding idea or goal for system design.

However a ground can only act as a sink or source for current when charges can accumulate, in other words when a capacitor with sufficient capacity is present.

Since $V=\frac Q C~$, $dV=\frac {dQ}{C}~$, If we want the ground potential, which is traditionally set to zero, to not change as the amount of charge changes, then the ground capacitance must approach infinity?

 

[amylostorm]

There may be some communication issues here. For example, for a given circuit diagram, the values of all resistors, capacitors, resistors and their connection methods are determined and cannot be changed, so of course you can choose any point as a ground when analyzing the circuit diagram. But I believe EE won't say that you can ground any point in any real circuit without changing the distribution of charge throughout the system, which is obviously impossible because the charge in the grounded conductor will be discharged or neutralized.

Sorry for the late reply, I sometimes forget to check my mail.

I do believe that EE says that grounding any point(just one though) will not change the distribution of charge through the system. I mean, we do that with the neutral wire in power distribution too. Every transformer has one end of its secondary loop connected to ground and it doesn't change anything about the circuit. Of course now that the neutral is at ground potential it does change some other things around. As in say I touch the neutral wire and if everything is working as intended I won't get shocked. If the circuit were floating, I would. We could also ground the "live wire" and then the dangerous one would be the neutral wire with a negative potential difference to ground...I think that was done with some early telegraph systems(though they were DC, but my point still stands).

I don't think that a grounded conductor is required to be neutral. It takes on the charge distribution that minimizes it's energy(I've finally understood @TSny 's argument!) which might or might not make it neutral. Could you explain why you think so?

I don't think it necessarily needs to be very far. We can take one conductor in the system as a ground reference, and all absolute voltages are measured relative to that conductor. Otherwise, we could also connect the conductor to a larger object, one with almost infinite capacitance. Then charge can freely removed or added to ground without changing its potential.

I agree with all of those statements, but I was actually trying to say that the potential of the ground is important in determining the final state of a charge distribution and that taking it as reference just means we can say that $V_{\infty} = -V$ where $V$ was the potential of the ground before. The final state now depends on the potential of $\infty$ w.r.t. our infinite coductor so it's still the same condition but written in a different form ie semantics.

 

[alan123hk]

Although if the earth is viewed as a conductive sphere, its capacitance is only 710uF, which seems pitifully small. But looking at it another way, considering the capacitance between the earth and the lower ionosphere, assuming an altitude of 50 kilometers, the capacitance between these concentric spheres is a rather large capacitor, about 90mF, but still does not seem to take into account the approximate infinity.

Although the capacitance of the earth is smaller than we imagine, the biggest advantage of choosing it as the ground is that its area is very large, so it provides convenience to all people living on the surface of the earth.

We can illustrate with an example. Suppose an alien object with a charge of 100C falls on the Earth. The charge on the Earth's surface increases by 100C. Then according to $~ dV=\frac{dQ}{4\pi\epsilon_0R}$, the potential at the Earth's surface relative to infinity will increase by 150kV

But if we only consider the potential changes in space one kilometer perpendicular to the Earth's surface relative to the Earth's surface, then according to formula $~\frac{Q_2-Q_1} {4 \pi \epsilon_0} \left( \frac{1}{R_2}-\frac{1}{R_1} \right)$, the potential change is only -25V, so It is conceivable that the impact on people living near the earth's surface or facilities built on the earth's surface will actually be small.

 

[amylostorm]

Interesting, I've never really thought about it like that, but it makes sense. Earth's radius is large but $K = \frac{1}{4 \pi \varepsilon_0} \approx 9\cdot10^9$ is larger so it's capacitance( $\frac R K$ ) is very small, which is completely opposite to our usual assumption of an infinite capacity ground. Amazing!

 

[alan123hk]

Every transformer has one end of its secondary loop connected to ground and it doesn't change anything about the circuit. Of course now that the neutral is at ground potential it does change some other things around. As in say I touch the neutral wire and if everything is working as intended I won't get shocked. If the circuit were floating, I would. We could also ground the "live wire" and then the dangerous one would be the neutral wire with a negative potential difference to ground...

As far as I know, we can connect one of the two secondary outputs of the isolation transformer to ground and of course the output voltage will not change in this case. But this involves safety issues that need to be handled very carefully. The reason is that one of the main functions of isolation transformer is to significantly reduce the risk of electric shock caused by high voltage between the two outputs and ground (But of course the high voltage between the two output ports is always present) . If one of the outputs is grounded (earthed) , the other output will develop dangerously high voltages to ground (earth).

I'm not an electrical engineer, so I don't know much about connect neutral to ground, but I think this is only allowed if it complies with relevant regulations and technical design, and is performed by a qualified electrical engineer. Where I live, the neutral and ground (earth) wires of the home electrical system should never be connected together. https://www.circuitsgallery.com/what-happens-if-you-connect-neutral-to-ground/

I don't think that a grounded conductor is required to be neutral. It takes on the charge distribution that minimizes it's energy(I've finally understood @TSny 's argument!) which might or might not make it neutral. Could you explain why you think so?

Yes, actually, the ground is not necessarily neutral. The ground may also have an electrical charge. The so-called grounding and neutralization of a conductor only means that its voltage becomes the same as the ground voltage, or the original excessively concentrated charge on the conductor combines with the original charge of the earth, and then the overall distribution is dispersed.

It's an interesting phenomenon where the charges in a system try to distribute themselves to achieve the minimum potential energy of the entire system. This is a bit like something placed high releasing potential energy when it falls to the ground, or like a slender spring releasing energy when it retracts.

An example can be used to illustrate that in space, the closer the distance between two identical charges, the greater the potential energy of the system, and the farther the distance, the smaller the potential energy of the system. On the contrary, the closer the distance between two opposite charges, the smaller the potential energy of the system, and the farther the distance, the greater the potential energy of the system.

So, in a parallel plate capacitor, if the charges on the two plates are the same, they will be concentrated on the outer surfaces of the two plates, whereas if the charges on the two plates are opposite, they will be concentrated on the inner surfaces of the two plates.