Ampere's Law when applied to a toroid

[Regtic]

It's a simple question:

Why is I_enc in Ampere's Law in a toroid equal to µ₀NI where N is the number of loops around the toroid? Why is I N times greater when the wire is looped around the toroid? When you take a random wire with current and you change the geometry of the wire into a loop, does the current actually increase? Shouldn't I_enc remain I?

 

[Matterwave]

I think you're confusing what a toroid is. A toroid is not a wire made into a loop, it's a solenoid made into a loop. It's a loop of loops!

Imagine a cylinder around which you wrap a wire, like string. And then imagine taking this cylinder and wrapping it around itself so that the two ends meet. Now you have a toroid. This is very different than a circular current loop, which is just a wire turned into a circle.

 

[Regtic]

No, I mean in the toroid apparatus, I guess. I know what a toroid is! I'm looking at a picture of it right now. I'm just wondering why the current would be NI for a toroid but in any other case the current is equal to V/R regardless of geometry. Also I'm just confused as to why it's NI in the first place.

 

[Regtic]

Oh my god I think I just figured it out by myself! Is it because I is defined as charge/second at a specific cross section so if we measure the current around a toroid we have to take into account the number of cross sections/loops?

But wait - aren't the cross sections of the wire different than the loops in the toroid?

 

[dauto]

It doesn't have to be a toroid. Look up solenoid and you will also see the fact NI. that N is just the number of loops.

 

[dauto]

Oh my god I think I just figured it out by myself! Is it because I is defined as charge/second at a specific cross section so if we measure the current around a toroid we have to take into account the number of cross sections/loops?

But wait - aren't the cross sections of the wire different than the loops in the toroid?

Why is it that when people are about to figure out the answer to their question they get confused by a red herring?

 

[jtbell]

In principle, one could construct a toroid or a solenoid using N separate loops of wire, driven by N separate power sources, each supplying current I, or in parallel from a single power source supplying total current NI. It wouldn't make any difference as far as Ampère's Law is concerned, versus the normal configuration with a single helically-wound wire carrying current I in N sequential loops.

 

[Regtic]

So say we took a wire and we measure that it has a current I with a resistance R connected to a power source with a voltage V.

Now, we take that same wire and wrap it around a cylindrical loop N times with the same power supply, it would still have a total current I, right? It's just that when we talk about toroids we usually refer to the current in each loop of wire as I with a total current NI instead of it having a total current I with current I/N in each loop, right? (Because we can construct a toroid with separate loops of wire rather than just one coiled wire)

 

[dauto]

So say we took a wire and we measure that it has a current I with a resistance R connected to a power source with a voltage V.

Now, we take that same wire and wrap it around a cylindrical loop N times with the same power supply, it would still have a total current I, right? It's just that when we talk about toroids we usually refer to the current in each loop of wire as I with a total current NI instead of it having a total current I with current I/N in each loop, right? (Because we can construct a toroid with separate loops of wire rather than just one coiled wire)

There is no change in convention. The current I in the wire before wrapping it around the toroid is the same current I after wrapping it up. The current used in Ampere's law is NI because each loop contributes an identical current I adding up to a total NI. It can't be any plainer than that.

 

[Regtic]

There is no change in convention. The current I in the wire before wrapping it around the toroid is the same current I after wrapping it up. The current used in Ampere's law is NI because each loop contributes an identical current I adding up to a total NI. It can't be any plainer than that.

So then how does the current get multiplied by N times if the voltage and resistance stay the same?

 

[dauto]

The current doesn't get multiplied by N. It is still I. Each loop has a current I. The total current going around the toroid is NI because the each loop contributes a current I and there are N loops.

 

[Regtic]

The current doesn't get multiplied by N. It is still I. Each loop has a current I. The total current going around the toroid is NI because the each loop contributes a current I and there are N loops.

So then if the current doesn't get multiplied by N, then the total current remains the same. It remains as I. So then why are we saying that each loop has current I making the total current NI if the total current should still be I as it was in the previous circuit? It is a change of convention, no? If we kept the convention, the current in each loop would be I/N.

 

[dauto]

So then if the current doesn't get multiplied by N, then the total current remains the same. It remains as I. So then why are we saying that each loop has current I making the total current NI if the total current should still be I as it was in the previous circuit? It is a change of convention, no? If we kept the convention, the current in each loop would be I/N.

Now you're tiring me. There is no change of convention. The current through each loop remains I. the total current around the toroid is NI because each loop contribute a current I and there are N loops. If you are watching a car race between I cars and each car goes around the track N times you will see NI cars going by because there are I cars but each car goes by N times. It's the same logic.

 

[Regtic]

Now you're tiring me. There is no change of convention. The current through each loop remains I. the total current around the toroid is NI because each loop contribute a current I and there are N loops.

But the current in each loop is just a fraction of the total current of the wire which we found was I when we set it up connected to a battery. So the current in each loop can't be I.

If you are watching a car race between I cars and each car goes around the track N times you will see NI cars going by because there are I cars but each car goes by N times. It's the same logic.

This analogy doesn't work. What is the car? A charge? Current is the number of charges passing through a cross section per second. I would be the number of cars you see pass by you per second. I is a rate, not just a number I'm not trying to tire you or annoy you, I'm going to thank you when you prove me wrong. I'm just not going to pretend I understand when I can point out why I don't agree with something.

 

[Regtic]

If a plane covers a distance x over time t, then the speed of the plane is said to be V.

If the plane then goes through a tunnel of distance x in time t that loops around a cylinder akin to the wire in the toroid, then the speed remains to be V, regardless of how many loops it goes through in the apparatus.

 

[dauto]

But the current in each loop is just a fraction of the total current of the wire which we found was I when we set it up connected to a battery. So the current in each loop can't be I.

No, all the current goes through each loop.

This analogy doesn't work. What is the car? A charge? Current is the number of charges passing through a cross section per second. I would be the number of cars you see pass by you per second. I is a rate, not just a number

Yes, that's right. The rate increases in a multiple lap race because you see the same car more than once.

I'm not trying to tire you or annoy you, I'm going to thank you when you prove me wrong. I'm just not going to pretend I understand when I can point out why I don't agree with something.

Fair enough

 

[dauto]

If a plane covers a distance x over time t, then the speed of the plane is said to be V.

If the plane then goes through a tunnel of distance x in time t that loops around a cylinder akin to the wire in the toroid, then the speed remains to be V, regardless of how many loops it goes through in the apparatus.

Yes, the speed of the plane doesn't change, but the number of planes per second increases because you're seeing the same plane go by more than once.

 

[Regtic]

No, all the current goes through each loop.


Yes, that's right. The rate increases in a multiple lap race because you see the same car more than once.

Fair enough

K so I'm still exhausted so maybe posting this wasn't a great idea but I'm going crazy over this.

If I =nqV_dA
where n is the number of mobile charge carriers per unit volume
q is the charge on each carrier
V_d is the speed of the charge carriers
A is equal to the area of the cross section

Then the current is independent of the geometry of the wire or the number of loops.

Another argument: If the current is proportional to the voltage of the battery, and the battery doesn't change, how come the total current becomes NI rather than I? You don't have N more identical batteries.

To clarify what I meant earlier

Current is measured in the amount of charge passing through a cross section of a wire per second in a circuit connected to a battery. Now if you take that wire and coil it around a cylindrical loop, the charge passing through any given cross section is still I, correct? So the current in the wire is I, and thus the current in toroid is I. I isn't the charge per second through a closed loop, it's the charge per second through the entire wire. That's why in your analogy, if there are I cars passing by per second, then the number of cars passing you per second is I. It doesn't make sense to say NI is the total current because the total current is the total amount of cars passing by you per second, which is I.

TL;DR Why is it that the current in each loop is I rather than the current in the entire wire is I?

I just feel terrible about writing all this when I know I'm wrong and making people read this nonsense but I really want to understand this before I go to sleep.

 

[Matterwave]

Sorry for my late response, but it looks like instead of a fundamental misunderstanding of the nature of a toroid, you have a fundamental misunderstanding of the nature of Ampere's law.

Ampere's law says:

$$\oint _C \vec{B}\cdot d\vec{l}=\mu_0\iint_S \vec{J}\cdot d\vec{S}\equiv\mu_0 I_{enc}$$

This is true for when the boundary of the surface S is the curve C.

What does this mathematical expression mean? It means that given a closed loop (called an Ampere loop) the integral of the magnetic field around this closed loop is equal to a constant times this quantity called $I_{enc}$ which is actually equal to the surface integral of the current density piercing the surface S with boundary C.

SO, we must define what curve C we are looking at, and therefore what surface S we are looking at. In a toroid, the curve C is taken to be a circle following along the inside of the toroid (by inside, we mean inside the cylinder we constructed before, not inside the donut hole, imagine the toroid as a donut, then the curve C is a circle inside the bread of the donut).

ANY surface with this curve C as the boundary will be pierced by the wire N separate times (you can convince yourself of this, by taking S to be simply the flat surface defined by the circle C). Therefore, we know that $I_{enc}=NI$. It is NOT that the current has somehow gone up when we turned a wire into a toroid. The current across the whole wire is I. But this wire PIERCES the surface S defined by the Ampere loop C a total of N times.

In other words $I_{enc}$ is NOT the current through the wire, but the total current that pierces the Ampere surface. In some examples (e.g. the straight wire) it just so happens that $I_{enc}$ is equal to the current carried by the wire but this is not always the case.

 

[Regtic]

Sorry for my late response, but it looks like instead of a fundamental misunderstanding of the nature of a toroid, you have a fundamental misunderstanding of the nature of Ampere's law.

Ampere's law says:

$$\oint _C \vec{B}\cdot d\vec{l}=\mu_0\iint_S \vec{J}\cdot d\vec{S}\equiv\mu_0 I_{enc}$$

This is true for when the boundary of the surface S is the curve C.

What does this mathematical expression mean? It means that given a closed loop (called an Ampere loop) the integral of the magnetic field around this closed loop is equal to a constant times this quantity called $I_{enc}$ which is actually equal to the surface integral of the current density piercing the surface S with boundary C.

SO, we must define what curve C we are looking at, and therefore what surface S we are looking at. In a toroid, the curve C is taken to be a circle following along the inside of the toroid (by inside, we mean inside the cylinder we constructed before, not inside the donut hole, imagine the toroid as a donut, then the curve C is a circle inside the bread of the donut).

ANY surface with this curve C as the boundary will be pierced by the wire N separate times (you can convince yourself of this, by taking S to be simply the flat surface defined by the circle C). Therefore, we know that $I_{enc}=NI$. It is NOT that the current has somehow gone up when we turned a wire into a toroid. The current across the whole wire is I. But this wire PIERCES the surface S defined by the Ampere loop C a total of N times.

In other words $I_{enc}$ is NOT the current through the wire, but the total current that pierces the Ampere surface. In some examples (e.g. the straight wire) it just so happens that $I_{enc}$ is equal to the current carried by the wire but this is not always the case.

THANK YOU! SO MANY THANKS! This is perfect!


If I'm honest, I spent like 4 hours just thinking about this today and you've cleared it up in an explanation that took me a minute to read. I'm very grateful that you took the time to explain that to me.

 

[Matterwave]

Glad to help. :)

 

[Philip Wood]

Current is measured in the amount of charge passing through a cross section of a wire per second

No it isn't. That's how the current through the wire is defined. But If we consider a plane passing through one side of the solenoid and containing the solenoid axis, the current through this plane will be NI.