What is the tension in a charged ring?

[etotheipi]

Homework Statement

The ring I'll assume to have negligible thickness, and linear charge density $\lambda$, as well as a radius $R$.

Relevant Equations

N/A

I tried considering a little piece of the ring (shaded black below) subtending angle $d\theta$, and attempted to find the electric field in the vicinity of that piece by a summation of contributions from the rest of the ring:

$$dE_x = \frac{dq}{4\pi \epsilon_0 d^2} \cos{\phi} = \frac{\lambda R d\theta}{4\pi \epsilon_0 \cdot 2R^2(1-\cos{\theta})}\cos{\phi}$$ $$E_x = \int_0^{2\pi} \frac{\lambda \sin{\frac{\theta}{2}}}{8\pi \epsilon_0 R(1-\cos{\theta})} d\theta = \frac{\lambda}{8\pi \epsilon_0 R} \int_0^{2\pi} \frac{\sin{\frac{\theta}{2}}}{1-\cos{\theta}} d\theta $$ Problem is, that thing diverges because the denominator goes to zero at the boundaries (the indefinite is easy enough to solve by changing $1-\cos{\theta} = 2\sin^2{\frac{\theta}{2}}$). If we call $$I = \int_{0}^{2\pi} \frac{\sin{\frac{\theta}{2}}}{1-\cos{\theta}} d\theta$$ then the electrostatic force on the piece in terms of this is $$F_e = (\lambda R d\theta) \frac{\lambda I}{8\pi \epsilon_0 R} = 2T\sin{\frac{d\theta}{2}} \approx T d\theta$$ and the tension would be, under the usual equilibrium constraint, $$T = \frac{\lambda^2 I}{8\pi \epsilon_0} = \frac{q^2 I}{32 \pi^3 R^2 \epsilon_0}$$but that doesn't make much sense if $I$ is infinite. I wondered if anyone could help out? Perhaps it is the case that the electric field diverges at the ring, but in that case, how could I modify my calculations to get a sensible answer? Because I am fairly sure that in reality the tension will not be infinite . Thanks!

 

[PeroK]

Can you explain why the assumed physical scenario may not be valid?

Hint: consider a finite set of charges.

 

[etotheipi]

Can you explain why the assumed physical scenario may not be valid?

Hint: consider a finite set of charges.

I'm not too sure. If you replace the homogenous linear charge with a bunch of individual point charges spaced at regular intervals, then the charge density $\rho$ will diverge at the point charges (since we can make the domain $V$ arbitrarily small), and be zero elsewhere. That will mean that $\vec{E}$ is undefined at the point charges.

It would seem that the electric field diverges here too, in the radial direction, at the ring.

 

[PeroK]

I'm not too sure. If you replace the homogenous linear charge with a bunch of individual point charges spaced at regular intervals, then the charge density $\rho$ will diverge at the point charges (since we can make the domain $V$ arbitrarily small), and be zero elsewhere. That will mean that $\vec{E}$ is undefined at the point charges.

It would seem that the electric field diverges here too, in the radial direction, at the ring.

Yes, you have two problems. Trying to calculate the field on a line charge and trying to calculate a tension on a perfect circle.

 

[etotheipi]

Yes, you have two problems. Trying to calculate the field on a line charge and trying to calculate a tension on a perfect circle.

I'm happy with the first part, in some cases like a 3D shell of charge we can get perfectly well defined electric fields at the boundary, but not so much for a line of charge, and evidently this example too.

But I wonder what is wrong with calculating tension on a perfect circle? A similar technique is used to look at mechanical capstans, or in 3D we could consider a small surface element of a soap bubble and derive an equilibrium equation to relate surface tension to the pressure differential.

 

[PeroK]

I'm happy with the first part, in some cases like a 3D shell of charge we can get perfectly well defined electric fields at the boundary, but not so much for a line of charge, and evidently this example too.

But I wonder what is wrong with calculating tension on a perfect circle? A similar technique is used to look at mechanical capstans, or in 3D we could consider a small surface element of a soap bubble and derive an equilibrium equation to relate surface tension to the pressure differential.

Perhaps you could get the two infinities to cancel out? What if you consider a small element on the right, and calculate the tension on that and take the limit as the size of the element goes to zero?

I don't know if that would work out. It's just an idea.

 

[etotheipi]

Perhaps you could get the two infinities to cancel out? What if you consider a small element on the right, and calculate the tension on that and take the limit as the size of the element goes to zero?

I don't know if that would work out. It's just an idea.

Actually that's a pretty good idea, have some (no longer infinitesimal) control region on the right that subtends an angle $\alpha$ or something, and try to come up for an expression for the electrostatic force on that.

It's slightly more complicated maths-wise and I'm not so sure how best to do it yet, but I think that would give something useful! Let me think about it for a while

Thanks!

 

[Adesh]

I'm happy with the first part, in some cases like a 3D shell of charge we can get perfectly well defined electric fields at the boundary, but not so much for a line of charge, and evidently this example too.

But I wonder what is wrong with calculating tension on a perfect circle? A similar technique is used to look at mechanical capstans, or in 3D we could consider a small surface element of a soap bubble and derive an equilibrium equation to relate surface tension to the pressure differential.

We cannot a get a good form for electric field at points of the circle beacuse the field is not radial. I doubt if there exist a nice mathematical form of electric field other than on the central axis.

Our (I mean yours) loop has to have some outward stretching for tension to exist by that external force, otherwise all the tension is due to the material of loop being bent to form a circle.

 

[haruspex]

Line charges are not physically valid. They only work as approximations of the field due to a wire at distances from the wire rather greater than its thickness. At the wire itself you get an infinite field.

You might be able to resolve it by considering the field at a point on the line as due to separate sections: a nearby region, treated as a straight hollow cylinder, and the rest treated as a zero thickness circular arc.
Figuring out where to do the cutover will be interesting.

Edit: no, can't assume it is ok to treat it as a straight cylinder.. the field would have no component in the direction of interest. Need some way to bound the component.

 

[etotheipi]

Line charges are not physically valid. They only work as approximations of the field due to a wire at distances from the wire rather greater than its thickness. At the wire itself you get an infinite field.

You might be able to resolve it by considering the field at a point on the line as due to separate sections: a nearby region, treated as a straight hollow cylinder, and the rest treated as a zero thickness circular arc.
Figuring out where to do the cutover will be interesting.

Thanks, I didn't know that was a general property.

Your suggestion is quite interesting, it seems like a very good idea. Like you mentioned, I am not sure how to judge where to split it up. I'll try and play around with it for a bit tomorrow morning!

I was also thinking, if the line charges are not physical, then might it be easier to model the loop with a torus of non-zero cross sectional area, probably as an insulator with uniform charge density to keep things simple? Maybe this will also be too complicated. Perhaps it is also possible to calculate the potential energy of such an arrangement, and from that derive some sort of electric force. I don't know; in any case, puzzles for tomorrow .

 

[Vanadium 50]

Why do you want to use calculus?

 

[etotheipi]

Why do you want to use calculus?

No particular reason, but it's the only way in I could see. Is there a better way?

For a 3D shell you can use superposition ideas because it's easy to find the electric field inside and outside, but here it doesn't seem as straightforward.

 

[Vanadium 50]

Yes...not calculus.

Can you replace the field from a ring of charge with that from a point? Can you determine the energy of a ring of charge using that? How does it depend on Δr? If the ring were also a spring, what would the change in its energy be as a function of Δr?

 

[etotheipi]

Yes...not calculus.

Can you replace the field from a ring of charge with that from a point? Can you determine the energy of a ring of charge using that? How does it depend on Δr? If the ring were also a spring, what would the change in its energy be as a function of Δr?

Interesting! Right now I probably can't because I am about to fall asleep, but a few things that come to mind before then...

It's not immediately obvious to me how to find the effective point charge, but I'm guessing it's going to be in the centre of the ring. I would have thought you obtain the self energy by then integrating up the energy density $\frac{1}{2}\epsilon_0 E^2$ over all space, but you said no calculus, so that can't be right...

I am not sure; a point charge would have a field strength that falls of as an inverse square with radius but the one I derived initially was only inversely proportional to the first power of $R$. Finding the point charge seems to be the key bit, though I'm a little stuck

In 3D you can find an outward pressure by differentiating w.r.t. volume, but here we can't do that. Maybe taking the derivative w.r.t. $r$ gets something with dimensions of force, but I don't know how to interpret this exactly?

Maybe you could replace the linear charge with $n$ point charges equally distributed, then the potential energy would be a summation involving $^n C_2$ terms each of which is a bit like $$\frac{q^2}{4 \pi \epsilon_0 \sqrt{2R^2(1-\cos{\frac{2\pi m}{n}})}}$$ perhaps $\frac{dU}{dr}$ is useful.

Maybe it will be more obvious in the morning

 

[Vanadium 50]

Google shell theorem.

 

[Delta2]

I was also thinking, if the line charges are not physical, then might it be easier to model the loop with a torus of non-zero cross sectional area, probably as an insulator with uniform charge density to keep things simple? Maybe this will also be too complicated.

That's what I was thinking from the first moment I saw this thread. Math will get complex but I think its the only nice way I see to get rid of the infinity of the electric field. Probably the end result will be something that has afunction of $(R_{in}-R_{out})$ in the denominator ($R_{in},R_{out}$ the inner and the outer radius of the torus), and it will also diverge if $R_{in}=R_{out}$ which is the case that the torus degenerates to a circle.

 

[haruspex]

it will also diverge if $R_{in}=R_{out}$ which is the case that the torus degenerates to a circle.

Not necessarily. There are two limit processes here, that involved in the integration and that in letting the radius of the wire tend to zero. Swapping the order so that the integration is done first might not lead to the same result.

 

[Adesh]

One thing that is coming to my mind constantly is if we have a steel wire (negligible thickness) then we know that it can stretched and tension can be developed in it by straining it. Now, imagine we charge the steel wire with continuous charge distribution, as all the charges are same they will repel each other and therefore we might be tempted to say there would exist a tension in the steel wire due to electric repulsion but when we go to mathematics for the calculation of repulsive force we find that charges are so close to each other that there will be an infinite repulsion on any segment of the wire.

So, how about first calculating the tension on a straight wire and then moving on to the circular loop.

 

[Delta2]

We ll run into similar problem trying to calculate the tension in a straight wire of infinitesimal thickness. The reason is that by Gauss's law in differential form we ll have that $$\nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}$$ and because the wire is of infinitesimal thickness the charge density $\rho$ will be a dirac delta function $$\rho(\mathbf{r})=\lambda\delta(\mathbf{r}-\mathbf{r_0}),\mathbf{r_0}\in L$$, hence $$\nabla\cdot\mathbf{E(r_0)}=\infty$$.

I believe that infinite divergence of the E-field in this case means that the E-field becomes infinite. In some other cases it means that the E-field does a jump between two finite values.

 

[Delta2]

Not necessarily. There are two limit processes here, that involved in the integration and that in letting the radius of the wire tend to zero. Swapping the order so that the integration is done first might not lead to the same result.

I don't agree, I think even if we do the integration first, we ll get something that will give infinities as $R_{in}\to R_{out}$

 

[Adesh]

Well we can think about this problem in an analogous way as drop of water. The pressure inside the drop is more and the pressure outside is less, so water drop should expand but it’s the surface tension that is preventing it from increasing its radius.

For the present case, we need to have a total force on the loop, which we can find by finding the electric force on a element and integrating all over the loop, then this total outward force will be equal to tension in the equilibrium case.

 

[etotheipi]

Google shell theorem.

I can prove the shell theorem for a shell, but not a ring. I think this problem is actually very easy in 3D, since the inside field is zero and field just outside the surface is $\frac{\sigma}{\epsilon_0}$. By the principle of superposition we immediately know that the field at the surface from the rest of the shell is finite and equal to $\frac{\sigma}{2\epsilon_0}$, from which we can get an internal electrostatic pressure (for instance, it is just the energy density ). Then we could finally consider a surface element acted upon by 4 tension forces and solve for equilibrium.

But I don't know of any shell theorem in the 2D case? I wonder if there is a generalisation we can make?

That's what I was thinking from the first moment I saw this thread. Math will get complex but I think its the only nice way I see to get rid of the infinity of the electric field. Probably the end result will be something that has afunction of $(R_{in}-R_{out})$ in the denominator ($R_{in},R_{out}$ the inner and the outer radius of the torus), and it will also diverge if $R_{in}=R_{out}$ which is the case that the torus degenerates to a circle.

I think I might try this. At the moment I'm not too sure how to go about finding any electric fields, but it's surely possible?

 

[Adesh]

We cannot get an explicit expression for the force at any point on the boundary of the loop, therefore we shall not be able to calculate the tension.

 

[haruspex]

I don't agree, I think even if we do the integration first, we ll get something that will give infinities as $R_{in}\to R_{out}$

Quite possibly, just saying you should not assume that.

 

[Vanadium 50]

But I don't know of any shell theorem in the 2D case

Did you look at the proof? (Hint: I already know the answer to that)

Normally I don't like providing complete (or near complete) solutions, but people keep pushing you into taking integrals.

Define r₁=R and r₂ = R+ΔR.
The ring is a spring of spring constant k, so the energy in it is 1/2 kx². The difference in energies as it goes from r₁=R to r₂ is Aπ²kRΔR, so the tension is Aπ²kR. A is completely calculable, but I leave calculating it as an exercise to the student.

The total charge is 2πRλ, Because of the shell theorem, I can treat this as coming from a single point at the center of the loop. The energy at point r₁ is [1/4πε₀]( 2πRλ )²/r₁ (do you see why? Particularly why 2πRλ is squared?), and at r₁ is [1/4πε₀]( 2πRλ )²/r₂. The difference between this is BΔR.

B is completely calculable, but I leave calculating it as an exercise to the student.

Consideration of energy means Aπ²kRΔR = BΔR, which let's you solve for k in terms of A and B. You plug that back into T = Aπ²kR.

You don't need to calculate electric fields, and you certainly don't need to be taking integrals hither and yon. You do need to understand the symmetries of the problem, and you have to understand the ideas behind the various rules and formulas, and not just memorize the outcome.

Finally, a few hours calculating can save you from a few minutes thinking.

 

[haruspex]

Because of the shell theorem, I can treat this as coming from a single point at the center of the loop.

THis is the part that seems to puzzle everyone else on the thread. As far as I know, that is only for spherical symmetry, which we do not have here.

 

[Vanadium 50]

That's why I said "look at the proof".
The 3-D shell theorem is proved by summing up rings.

 

[etotheipi]

Define r₁=R and r₂ = R+ΔR.
The ring is a spring of spring constant k, so the energy in it is 1/2 kx². The difference in energies as it goes from r₁=R to r₂ is Aπ²kRΔR, so the tension is Aπ²kR. A is completely calculable, but I leave calculating it as an exercise to the student.

The total charge is 2πRλ, Because of the shell theorem, I can treat this as coming from a single point at the center of the loop. The energy at point r₁ is [1/4πε₀]( 2πRλ )²/r₁ (do you see why? Particularly why 2πRλ is squared?), and at r₁ is [1/4πε₀]( 2πRλ )²/r₂. The difference between this is BΔR.

B is completely calculable, but I leave calculating it as an exercise to the student.

Consideration of energy means Aπ²kRΔR = BΔR, which let's you solve for k in terms of A and B. You plug that back into T = Aπ²kR.

There are a few things here which I do not follow. Okay so our ring is a spring, if it's natural length is zero (?) then the increment in energy when undergoing an increment of $\Delta R$ is $$\Delta E = \frac{4\pi^2 k}{2}(R^2 + 2R\Delta R + \Delta R^2 - R^2) \approx 4\pi^2 k R \Delta R$$ Then you identify tension as $4 \pi^2 k R$. This is the first bit I don't understand. Sure, the work done by the force of tension is the negated increment of the elastic potential energy, however here the tension forces do not act in the radial direction, so obtaining the force from this work doesn't seem as straightforward as division by $\Delta R$.

Next you say we can treat the charge as being in the centre of the ring; But here there is no sufficient spherical symmetry for the shell theorem to apply (in fact, this would only work if the electric force obeyed an inverse first power law).

But even then, I can't tell what you are computing the energy of. The equation $\frac{(2\pi R \lambda)^2}{4\pi \epsilon_0 r}$ would be the electric potential energy of a point charge of charge $2\pi R \lambda$ and a ring of charge density $\lambda$ at a radius $R$ around the point charge, or indeed alternatively two such point charges. But then I thought you replaced the ring with a single point charge, so where does the other body in the pair come from?

I might miss your point but currently it seems there are too many missing pieces.

 

[haruspex]

That's why I said "look at the proof".
The 3-D shell theorem is proved by summing up rings.

Sure, but isn't the point at which the field is being considered on the axis of the ring in that proof? Here it is on the perimeter.

 

[Vanadium 50]

If you don't want to follow the proof, here's the heuristic. In the plane of the ring, field lines are radial by symmetry. I can always replace the ring by one a factor of 2 (or 10, or a million) smaller without changing the field strength anywhere it wasn't zero. (Obviously points that were inside and are now outside see different fields). Do that as many times as you want and you get your ring arbitrarily close to a point.

 

[Vanadium 50]

Okay so our ring is a spring, if it's natural length is zero (?)

Natural length is irrelevant. I only look at energy differences. Don't make the problem harder than it has to be.

however here the tension forces do not act in the radial direction

Did I talk about directions at all? You have a spring of one length and I change it's length, so I know how much energy it took. Don't make the problem harder than it has to be.

But here there is no sufficient spherical symmetry for the shell theorem to apply

You aren't going to look at the proof, are you? You will see the proof relies on a lemma involving rings.

But then I thought you replaced the ring with a single point charge,

When in 3D you use the shell theorem on a sphere, does its energy become infinite?

You want to use forces to solve this problem. You should energies.

 

[etotheipi]

Natural length is irrelevant. I only look at energy differences. Don't make the problem harder than it has to be.

I think the energy of a spring is $\frac{1}{2}k \delta^2 = \frac{1}{2}k(l-l_0)^2$ so for an infinitesimal change in length $dE = k(l-l_0)dl = k(2\pi r - l_0) \cdot 2\pi dr = 4\pi^2 k r dr - 2 \pi k l_0 dr$. The second term is dependent on the natural length, but you have dropped it.

Did I talk about directions at all? You have a spring of one length and I change it's length, so I know how much energy it took. Don't make the problem harder than it has to be.

Then how did you calculate the tension? For a straight spring being extended, we could say that $T dR = dE$ if $dR$ is the displacement of the end of the spring, however in this scenario the tension is not obtained by dividing the increment in energy by the change in radius, because the tension forces doing the work are not in the radial direction?

You aren't going to look at the proof, are you? You will see the proof relies on a lemma involving rings.

I have proved it myself earlier, and I did so by integrating up the electric field due to successive rings in the axial direction of those rings. What lemma are you referring to?

When in 3D you use the shell theorem on a sphere, does its energy become infinite?

You want to use forces to solve this problem. You should energies.

I don't know what you mean; the self energy of a sphere can be obtained by integrating $\int \frac{1}{2}\epsilon_0 E^2 dV$ and it is not infinite.

I have thought about using energies but I cannot see a good way of doing so .

 

[Vanadium 50]

The second term is dependent on the natural length, but you have dropped it.

True, but what does that term mean? It means that if there were no charge, that the ring was already under tension (or compression). The way I have defined it, the tension is zero if the charge density is zero. Had I got an answer like T = a + bλ² (you see why it needs to be squared, right?) I would associate a with the initial tension. You shouldn't add in an extra "intrinsic" tension (or compression). Don't make this harder than it has to be.

What lemma are you referring to?

The field contribution of each ring. Which is (in the plane of the ring) $[\frac{1}{4\pi\epsilon_0}]\frac{q}{r^2}$ outside, Directed radially. Inside the ring there is a divergence at r=R, so that can't be a solution. Don't make this harder than it has to be.

 

[etotheipi]

The field contribution of each ring. Which is (in the plane of the ring) $[\frac{1}{4\pi\epsilon_0}]\frac{q}{r^2}$ outside, Directed radially. Inside the ring there is a divergence at r=R, so that can't be a solution. Don't make this harder than it has to be.

Please excuse me if I make a mistake, but I constructed the contribution to the axial electric field from anyone ring as $$dE_r = \frac{Q}{16 \pi \epsilon_0 r^2 R} \frac{r^2 + c^2 - R^2}{c} dc$$ where $c$ is the length of the arm from any point on the ring to the place where we are evaluating the electric field. You say your expression is for the plane of the ring, but I don't know why you need this for the shell theorem.

(N.B. The whole result follows much more nicely from the theorem of Gauss, though)

But nonetheless, in the plane of the ring, outside the ring, I don't think that the field strength is $\frac{q}{4\pi \epsilon_0 r^2}$. How did you obtain that result?

 

[Vanadium 50]

How did you obtain that result?

Draw the field lines (which will be radial) outside the ring. Make the ring smaller. Do I have any more field lines? If so, where did they come from? Do I have less? If so, where did they go? Continue until the ring is a point.

 

[etotheipi]

Draw the field lines (which will be radial) outside the ring. Make the ring smaller. Do I have any more field lines? If so, where did they come from? Do I have less? If so, where did they go? Continue until the ring is a point.

It works for a 3D sphere, yes, but not for a 2D ring. There is not the necessary symmetry of field lines, we cannot yet tell whether the field strength is uniform in all directions.

For a 3D sphere, spherical symmetry implies uniform density of field lines emanating from the sphere. Theorem of Gauss tells me that, with a suitable spherical domain of integration $$\int_S \vec{E} \cdot d\vec{A} = 4\pi r^2 E= \frac{Q}{\epsilon_0}$$ and hence $E = \frac{Q}{4\pi \epsilon_0 r^2}$. But now let us consider a 2D ring. What surface are you integrating over? What symmetry do you invoke to ensure uniformity of field lines over this surface?

 

[Vanadium 50]

DRAW THE FIELD LINES.

Don't just tell me it doesn't work so you don't have to draw them. I'm trying to teach you something here, but I can't if you don't do anything I suggest. But I give up. Do it whatever way you want.

 

[etotheipi]

To the best of my knowledge, here is what I imagine the electric field to look like

black denotes the plane of the ring, red are a few extra outside of the plane of the ring. I missed out a few in the axial directions because I thought it would make the diagram unnecessarily hard to read. Now I construct an arbitrary control volume $V$ that encloses the ring. I can use the integral form of Gauss' law again, $$\int_S \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0}$$but I don't know how you can extract $E$ from that, because there is only a radial symmetry, and not a spherical symmetry like in the previous case.

I apologise if I upset you @Vanadium 50, but I can assure you I ask out of interest and not stubbornness . I appreciate your help. After all, you have the PhD and I do not

 

[TSny]

If the tension in the ring is finite, then the tension is equal to 1/2 of the force $F$ that one half of the ring exerts on the other half. For the case where the thickness of the wire (or thread) is zero, the tension will not be finite. But, I thought it would be interesting to put a couple of gaps in the ring as shown below and consider how the force $F$ varies with the size of the gap, $S$. $S$ is the arc length of each missing piece of the ring.

The force $F$ can be found by integration and I got the following result. For convenience, let $z$ denote the size of the gap in relation to the circumference of the ring: $z = \frac{S}{2 \pi R}$.

$$F = 2k\lambda^2 \left[ \cos(\pi z) \log \left[ \cot \left(\frac{\pi z}{2} \right) \right] + \sin (\pi z) - 1 \right ]$$
$k$ is Coulomb's constant. I think this is valid for any nonzero gap size; that is, for any $z$ in the range $0<z<\frac{1}{2}$. (As $z$ approaches $\frac{1}{2}$, the gaps consume the whole ring.)

Here are some numbers for a ring of circumference 1 meter and $k \lambda^2=1$N. (So, $\lambda$ is about 10 μC/m.) The gap is varied from 10 cm to 1 × 10^-13 m, which is much smaller than the size of an atom!

The force increases surprisingly slowly as the gap size decreases. For $z << 1$, the formula for $F$ can be approximated as $$F = 2 k \lambda^2 \left[ \log \left(\frac{2}{\pi z} \right) - 1 \right]$$

This shows how the force diverges logarithmically as the gaps shrink to zero.

 

[etotheipi]

The force increases surprisingly slowly as the gap size decreases. For $z << 1$, the formula for $F$ can be approximated as $$F = 2 k \lambda^2 \left[ \log \left(\frac{2}{\pi z} \right) - 1 \right]$$This shows how the force diverges logarithmically as the gaps shrink to zero.

This is very cool, thanks a bunch for the analysis! It's quite fun to play around with the formulae you provided.

I think I am probably quite (very) out of my depth to replicate the result, but I will have a go .

Thanks!

 

[haruspex]

If you don't want to follow the proof, here's the heuristic. In the plane of the ring, field lines are radial by symmetry. I can always replace the ring by one a factor of 2 (or 10, or a million) smaller without changing the field strength anywhere it wasn't zero. (Obviously points that were inside and are now outside see different fields). Do that as many times as you want and you get your ring arbitrarily close to a point.

That only says it is radial, so yes of course it is equivalent to some charge at the central point. I did not dispute that, and I would hope it was obvious to us all. The question is, what charge at the central point? Certainly not the total charge on the ring, as you seemed to imply by referring to the shell theorem without qualification.

Edit: indeed, looking back, I see you specifically stated as much:

The total charge is 2πRλ, Because of the shell theorem, I can treat this as coming from a single point at the center of the loop.

In particular, is it even finite? Everyone else in the thread was coming to the view that it probably was not, and I see @TSny has now confirmed that.

So to get a useful answer to the original question, the ring must be modeled as a torus, at least locally to the "test charge" point.

 

[etotheipi]

That only says it is radial, so yes of course it is equivalent to some charge at the central point.

But it is only equivalent in the plane, right? As far as I can tell there's no telling whether the field above the ring is also like that from a point charge, but my spidey-senses tell me that it's probably not. And that is why we can't deduce even the electric field outside the ring in the plane of the ring by heuristic arguments alone, because the problem lacks spherical symmetry.

I haven't made much progress with the torus today, but I will try harder tomorrow

 

[haruspex]

But it is only equivalent in the plane, right? As far as I can tell there's no telling whether the field above the ring is also like that from a point charge, but my spidey-senses tell me that it's probably not. And that is why we can't deduce even the electric field outside the ring in the plane of the ring by heuristic arguments alone, because the problem lacks spherical symmetry.

Agreed. It will not even seem like a point source when sensed across some small region. What I mean by that is that since the equipotential surfaces are not spheres they will locally have two different radii.
There's an interesting question... at a point on the ring, what is the curvature of the equipotential in a plane normal to the ring and through its centre?

 

[etotheipi]

I downloaded this wolfram project and apparently this is what the electric field looks like around such a charged ring:

It looks like we can vaguely imagine various equipotential surfaces (easier on the left diagram) by tracing through perpendicular to the field lines, as a first "approximation" .

 

[haruspex]

I downloaded this wolfram project and apparently this is what the electric field looks like around such a charged ring:

View attachment 265446View attachment 265447

It looks like we can vaguely imagine various equipotential surfaces (easier on the left diagram) by tracing through perpendicular to the field lines, as a first "approximation" .

Yes, it dawned on me that in the vicinity of the ring the equipotentials will be toroidal shells around the ring.

 

[Vanadium 50]

@etotheipi asked me to come back. Let me make a few comments:

People are, often implicitly, interpreting "negligible thickness" as "the ring is a torus, and we are taking the limit as the minor axis goes to zero." That reads more into the problem than is there. Further, they are discovering infinities when they do this. There are. The same infinities come up with any "line of charge" problem. Indeed, these infinities even come up when imagining a test charge as a tiny sphere of charge Q and letting the sphere radius go to zero. So to solve this, we take a "line of charge of charge density λ" at its word and...

It is a mistake to set up a problem so that the very elements of that problem are infinite.

So, I'm am taking the problem at its word - "negligible thickness" means treat this as a 2D problem, not as a 3D problem where I do not neglect the thickness. If you want to complain that this is not physically realizable, you can get in line with people complaining about frictionless surfaces, stretchless ropes, perfect spheres, etc.

Next, let's take a look at the picture in #38. The red lines are out of the plane and therefore irrelevant. I count six field lines in the plane. At 2R, 3R, 100R, I always have six lines. Everywhere outside the ring, the field configuration configuration looks like all of the charge is concentrated at a tiny point in the center of the ring.

Why take the outside field? Because it is finite, and presumably the author of the problem is looking for a finite answer.

OK, so what is the potential energy of this configuration? Let's look at a tiny slice of wire - say 1/1000th of it. It sees the field as if all the charge were at the center, and it's R units away. So it's [1/1000][1/4πε₀][Q²/R]. There are 1000 such pieces, so the total is [1/4πε₀][Q²/R]. A million pieces? Same answer. A trillion? Same answer.

So the energy change of moving the circle from R to R + ΔR is [1/4πε₀][(2πλ)²ΔR/R²]

The energy it takes to stretch a spring from R to ΔR is 4π²kRΔR.

Equating,

4\pi^2kR\Delta R = \frac{1}{4\pi\epsilon_0}\frac{(2\pi R \lambda)^2}{R^2}\Delta R

or

k = \frac{1}{4\pi\epsilon_0}\frac{\lambda^2}{R}

and so (if I didn't make an algebra error)

T = 4\pi^2 k R = \frac{\pi}{\epsilon_0}\lambda^2

 

[haruspex]

It sees the field as if all the charge were at the center

Please explain how you arrive at that. We all accept that by symmetry the net field is radial, but do not understand how you get that it is equivalent to the actual charge on the ring relocated to the centre (as it would be for a sphere).
In particular, you seem by this to magic away an infinity (or, equivalently, a critical dependence on the wire thickness) which provably exists.

 

[Delta2]

It sees the field as if all the charge were at the center

First of all only the field at the plane of the ring looks like as all the charge was at the center of the ring, but then again this is not accurate because we know that the field at the infinitesimally thin ring becomes infinite, while if we take the charge to be concentrated at the center, then the field at the thin ring is not infinite but has a finite value, it is right at the center that the field becomes infinite. So at first glance you move the infinity from the place of the thin ring to the place of center of the ring, so the two configurations do not seem to be equivalent (as they are equivalent in the case of a charged sphere with surface charge density, because we have spherical symmetry there).

 

[Vanadium 50]

but do not understand how you get that it is equivalent to the actual charge on the ring relocated to the centre (as it would be for a sphere).

Go back to counting field lines. If I have six at 10R and 2R, if I replace the ring with a smaller one - say R/4 - with the same change, I will still have six at 10R and 2R. And at R and R/2. Repeat as necessary.

while if we take the charge to be concentrated at the center,

There is no "center" to a ring of zero thickness (and/or height).

 

[Vanadium 50]

equivalently, a critical dependence on the wire thickness) which provably exists.

Yup. And why am I doing this? Because the problem said to.