Coulomb's Law and Conditional Convergent Alternating Harmonic Series

[plasticstardust]

Homework Statement

Mary Boas in her Mathematical Methods in Physical science Chapter 1 Section 8 has a discussion of conditionally convergent series that really perplexes me. Boas sets up an infinite line of discrete unit charges along the positive axial direction say +x with non linear spacing, and with signs on charges alternating such that x=0 is positive, x= 1 negative, x=\sqrt{2} positive, x=\sqrt{3} negative and so on. She calculates the forces on the unit positive charge at x=0 from all other charges producing the alternating harmonic series, which I concur with but then troublingly points out that since this series is conditionally convergent, the order in which terms are summed effects the convergence and sum. Surely, a physical observable cannot depend upon the order we calculate in.

Relevant Equations

F = Kqq/r^2

Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series anyway but as we approach the final placement the sum will look more and more like the sum of the alternating harmonic series.

I am unconvinced. The Net Force is calculated only at an instant, and doesn't reflect or incorporate any past history of how charges were placed. The ordering that produced the alternating harmonic series was artificial and an artifact of the calculation which I was inclined naturally to do as well in considering the example, before reading her solution. I do not believe that an artifact of the calculation method can effect the physical observable nor, but I do believe the result produced by coulombs law is conditionally convergent. I am unsure how to reconcile this.

My attempt at an explanation is that perhaps this a result of coulombs law being incorrect, with the correct explanation being given by some field theory incorporating propagation delay. Perhaps coulombs law isn't generally covariant, or dependant on geometry in some way that is forbidden to physical laws, and this is a failure of coulombs law to account for things like causality.

An alternative guess, is that perhaps this is an artifact of the situation bieng aphysical. No infinite line of point charges could ever occur in universe with finite charges. I am suspicious of this because infinite charge distributions in charges are used all the time and the results work out.

 

[PeroK]

An alternative guess, is that perhaps this is an artifact of the situation bieng aphysical. No infinite line of point charges could ever occur in universe with finite charges. I am suspicious of this because infinite charge distributions in charges are used all the time and the results work out.

That's what I'd say the answer is. An infinite sum is an approximation(!) to a physically finite sum. It may make it easier to calculate if we take the sum all the way to infinity.

In the example Boas provides, the physical scenario must be one of physically alternating charges up to some finite number. That is what we should calculate. This is approximated by an infinite line of alternating charges.

Although some results may be derived for infinite charge distributions, you always have to be careful that this result is an approximation of a large but finite configuration.

 

[plasticstardust]

That's what I'd say the answer is. An infinite sum is an approximation(!) to a physically finite sum. It may make it easier to calculate if we take the sum all the way to infinity.

In the example Boas provides, the physical scenario must be one of physically alternating charges up to some finite number. That is what we should calculate. This is approximated by an infinite line of alternating charges.

Although some results may be derived for infinite charge distributions, you always have to be careful that this result is an approximation of a large but finite configuration.

Thank you for your answer.

The ordering of the terms effects the ultimate sum and convergence behavior. Had I choose a different order to compute in say, net postive + net negative, or two positive charges then a negative, no convergent answer would appear.

In cases of conditional convergence like this, is it fair to say no answer is sensible because the problem isn't well defined? Or does this conditionally convergent series well approximate some physical reality? If a conditionally convergent series appears can the problem model reality? Physics doesn't seem to provide an order to sum terms in. In approximating reality with infinite charge distributions, what criteria do I have for selecting term ordering if anything with conditional convergence isn't excluded as not well-defined?

 

[PeroK]

Thank you for your answer.

The ordering of the terms effects the ultimate sum and convergence behavior. Had I choose a different order to compute in say, net postive + net negative, or two positive charges then a negative, no convergent answer would appear.

In cases of conditional convergence like this, is it fair to say no answer is sensible because the problem isn't well defined? Or does this conditionally convergent series well approximate some physical reality? If a conditionally convergent series appears can the problem model reality? Physics doesn't seem to provide an order to sum terms in. In approximating reality with infinite charge distributions, what criteria do I have for selecting term ordering if anything with conditional convergence isn't excluded as not well-defined?

The key idea is that you must take "partial sums" up to some point. The limit of an infinite series is then the limit of the partial sums. That is how the limit of an infinite series is defined. You cannot mess about with that! That then links the (correct) mathematical analysis approach with the idea of a physical approximation.

That is, by definition, the limiting process. If you don't follow the method of partial sums, then you're mathemtically off track.

Note that the commutative law, $a + b = b + a$, extends (by induction) only to finite sums not to infinite series. What Boas could have said is that the whole idea of swapping the order of summation is in general invalid for an infinite series. Although, you can prove that it is valid for absolutely convergent series.

The mathematical difficulty starts, therefore, as soon as you rearrange an infinite number of terms in an infinite series. It's as invalid as reversing the order of matrix multiplication, unless you have checked that your matrices commute. It's fundamental.

 

[plasticstardust]

The mathematical difficulty starts, therefore, as soon as you rearrange an infinite number of terms in an infinite series. It's as invalid as reversing the order of matrix multiplication, unless you have checked that your matrices commute. It's fundamental.

Thanks again for taking the time to think this through with me.

What order do you apriori decide to add charges in? If there are charges at all the locations mentioned above, and each contributes a Force to the charge on the origin, how is it that you decide which order to compute the sum in? To me it seems that there is No objectively correct, "physical"/"natural" ordering to sum the component forces in.

This would be less disturbing if all situations that produce conditionally convergent series are not well defined and aphysical. If they are approximations for physical reality but terms must be sumed in some special order , that seems be unreasonable to me.

 

[PeroK]

Thanks again for taking the time to think this through with me.

What order do you apriori decide to add charges in? If there are charges at all the locations mentioned above, and each contributes a Force to the charge on the origin, how is it that you decide which order to compute the sum in? To me it seems that there is No objectively correct, "physical"/"natural" ordering to sum the component forces in.

This would be less disturbing if all situations that produce conditionally convergent series are not well defined and aphysical. If they are approximations for physical reality but terms must be sumed in some special order , that seems be unreasonable to me.

If there are a finite number of charges, then you can add them in any order. If there are an infinite number of charges, then you can only add them (at all) by appealing to rigorous mathematical analysis and - once you have done that - you must pay attention to the precise definitions and theorems of mathematical analysis.

An infinite line of charges (re Coulomb's law) has no meaning without rigorous maths. If you want to consider an infinite line of charge, you must observe the results from mathematical analysis.

To be precise. 1) From the axioms of number theory we have:
$$\sum_{n =1}^N (a_n + b_n) = \sum_{n =1}^N a_n + \sum_{n =1}^N b_n$$
2) By definition:
$$\sum_{n =1}^{\infty} (a_n + b_n) = \lim_{N \rightarrow \infty} \sum_{n =1}^{N} (a_n + b_n)$$
3) In general:
$$\sum_{n =1}^{\infty} (a_n + b_n) \ne \sum_{n =1}^{\infty} a_n + \sum_{n =1}^{\infty} b_n$$

 

[plasticstardust]

If there are a finite number of charges, then you can add them in any order. If there are an infinite number of charges, then you can only add them (at all) by appealing to rigorous mathematical analysis and - once you have done that - you must pay attention to the precise definitions and theorems of mathematical analysis.

An infinite line of charges (re Coulomb's law) has no meaning without rigorous maths. If you want to consider an infinite line of charge, you must observe the results from mathematical analysis.

To be precise. 1) From the axioms of number theory we have:
$$\sum_{n =1}^N (a_n + b_n) = \sum_{n =1}^N a_n + \sum_{n =1}^N b_n$$
2) By definition:
$$\sum_{n =1}^{\infty} (a_n + b_n) = \lim_{N \rightarrow \infty} \sum_{n =1}^{N} (a_n + b_n)$$
3) In general:
$$\sum_{n =1}^{\infty} (a_n + b_n) \ne \sum_{n =1}^{\infty} a_n + \sum_{n =1}^{\infty} b_n$$

I understand, as you are pointing out, that with conditionally convergent series, you cannot reorder terms and this difficulty is caused by the infinite nature of the sum. The ordering is essential to the mathematics. What I do not understand is how you map a charge distribution to a ordering in a sum if order matters. Your pointing out that I cannot rearrange the order once an order is set. I am questioning how to map a charge distribution to an order in a sum in the first place.

 

[PeroK]

I understand, as you are pointing out, that with conditionally convergent series, you cannot reorder terms and this difficulty is caused by the infinite nature of the sum. The ordering is essential to the mathematics. What I do not understand is how you map a charge distribution to a ordering in a sum if order matters. Your pointing out that I cannot rearrange the order once an order is set. I am questioning how to map a charge distribution to an order in a sum in the first place.

You have to ask the following questions:

1) What physical scenario am I modelling with an infinite line of alternating charges?

Let's say that the answer is a long (but finite) line of alternating charge.

2) If the line is of length $L$ what would the total force be?

To do that you need an (approximately) equal number of plus and minus charges. The point is that the $\pm 1$ becomes negligible as $L$ increases.

3) Is $L$ long enough so that I can approximate my finite line charge by an infinite line charge?

If yes, then we can take the limit as $L \rightarrow \infty$.

What you can't do is take $L = \infty$ and then try to wing it and pretend that mathematical rigour doesn't matter. Note that $\infty$ is not a number or a valid length. So, you can't have that. You can only have $L \rightarrow \infty$. You must, therefore, set up a valid limiting process that models the physical scenario.

 

[plasticstardust]

You have to ask the following questions:

1) What physical scenario am I modelling with an infinite line of alternating charges?

Let's say that the answer is a long (but finite) line of alternating charge.

2) If the line is of length $L$ what would the total force be?

To do that you need an (approximately) equal number of plus and minus charges. The point is that the $\pm 1$ becomes negligible as $L$ increases.

3) Is $L$ long enough so that I can approximate my finite line charge by an infinite line charge?

If yes, then we can take the limit as $L \rightarrow \infty$.

What you can't do is take $L = \infty$ and then try to wing it and pretend that mathematical rigour doesn't matter. Note that $\infty$ is not a number or a valid length. So, you can't have that. You can only have $L \rightarrow \infty$. You must, therefore, set up a valid limiting process that models the physical scenario.

I think this might answer the question. If I understand you correctly, the situation proposed by Boas doesn't really model anything at all. Its aphysical to start with infinite discrete charges at infinite length. So, we get this conditionally convergent sum and have questions about how it was produced, but how it was produced doesn't matter because its arealistic to start.

I notice that the process of using continuous infinite distributions with standard definitions of integrals as limited sums does imply an ordering of the sum. But choosing a different coordinate system would result in "different ordering" when computing the sum that defines the integral and the answers should always be sensible.

Does this mean that physical, well defined problems cannot produce conditionally convergent sums? and that Boas' proposed problem is aphysical? Or am I misunderstanding?

 

[PeroK]

Does this mean that physical, well defined problems cannot produce conditionally convergent sums? and that Boas' proposed problem is aphysical? Or am I misunderstanding?

The scenario is, of course, not strictly physical. But that's not the point. More generally, if you imagine an infinite something that has inherent ambiguity. What you need is a well-defined limiting process. In this case, if the limiting process involves a finite line of charge that is neutral overall, then that is the approximation to some physical problem.

A variation of this would be to take positive charges out to a distance $L$ and negative charges out to a distance $2L$. Now the limit scenario (physically and informally) is the same as the first case: it's just the same infinite line of alternating charges. But, what you have actually modeled here is a long line of charges, where the negative charges extend twice as far as the positive ones.

This difference is not apparent from the final (aphysical) infinite configuration. It's the limiting process that defines the physical scenario being modeled; not the ambiguous limiting configuration.

 

[plasticstardust]

The scenario is, of course, not strictly physical. But that's not the point. More generally, if you imagine an infinite something that has inherent ambiguity. What you need is a well-defined limiting process. In this case, if the limiting process involves a finite line of charge that is neutral overall, then that is the approximation to some physical problem.

A variation of this would be to take positive charges out to a distance $L$ and negative charges out to a distance $2L$. Now the limit scenario (physically and informally) is the same as the first case: it's just the same infinite line of alternating charges. But, what you have actually modeled here is a long line of charges, where the negative charges extend twice as far as the positive ones.

This difference is not apparent from the final (aphysical) infinite configuration. It's the limiting process that defines the physical scenario being modeled; not the ambiguous limiting configuration.

I think I am starting to understand.

So, a limiting process defines the ordering of an infinite sum which may be conditionally convergent unambiguously because the order is defined.

Taking your example limiting process of placing charges out to L, but then deciding or order the (finite) sum out to L of charges by adding two positive charges then a negative and letting L \to \infty results in a sum that diverges problematic? Or would be the guidence that I always choose a limiting process that results in convergent sums?

 

[PeroK]

I think I am starting to understand.

So, a limiting process defines the ordering of an infinite sum which may be conditionally convergent unambiguously because the order is defined.

Taking your example limiting process of placing charges out to L, but then deciding or order the (finite) sum out to L of charges by adding two positive charges then a negative and letting L \to \infty results in a sum that diverges problematic? Or would be the guidence that I always choose a limiting process that results in convergent sums?

I'm going to quote Mary Boas. She started by saying: Here is a physical example of such a series which emphasizes the care needed in applying mathematical approximations in physical problems.

I can't say it any better than that.