2 hours later…
8:14 AM
@Charlie the proper distance is a calculated quantity not a measured one. Only a timelike trajectory allows a proper distance to be actually measured (on a clock carried along the path between the two points).
A spacelike trajectory between two points still has a meaning and can be calculated, but it could never be measured by an experiment.

3 hours later…
10:51 AM
in Mathematics, 4 mins ago, by Secret
but why does that work, why does expressing an $\Bbb{R}^2$ rotation in $\Bbb{C}^2$ and the two extra components can always cancel out in conjugate other than algebraic reasons, so one never end up with any leftover imaginary stuff?
had been revisiting really elementary stuff today for some reason

4 hours later…
2:34 PM
@JohnRennie Would it be valid to think of the proper distance then as the interval between two points on a timeslice? In other words if I effectively froze time and measured the distance between two events that's the "proper distance"? Which seems mathematically equivalent to setting $\text dt=0$ in the metric.
Sorry I hope you don't mind being pinged while you're not here I didn't check

Yes, though of course the events would be simultaneous only in one inertial frame.

I see, does the problem of finding a frame in which two events are simultaneous not become a problem in GR? Is it always possible to find a coordinate system in which two events are simultaneous?
Actually I think the answer to that is no, I should have added the qualifier "two timelike separated events"

No. The proper time is an invarient, so if the interval is timelike in one frame it is timelike in all frames.
If (and only if) an interval is spacelike then it is always possible to find a frame in which the two events are simultaneous.

@JohnRennie: Sir, I have a simple conceptual question.

Suppose we have a simple circuit with an ideal battery connected to resistance with ideal wires. We know that the potential difference across the resistance will be equal to "I*R", where I is the current and R is the resistance.

The question is, why is there a potential difference across the resistance? Is it due to the charge accumulation across the resistance? Is it true that the charge accumulation across the resistance will be such that the potential difference developed by it across the resistance will be equal to the potential d

Ok I think I'm getting it through to it, thank you John :)

2:46 PM
@DevanshMittal I've just started lunch, but I'll answer as soon as I'm finished.

@JohnRennie: Sure Sir. Thanks.
In the meantime, I will post some more point related to the same.

@Charlie In GR, the notion of "frame" isn't as simple as in SR. For your time slices to work you need to assume that the spacetime manifold can even be foliated properly by time slices (this property is called "globally hyperbolic" and the time slices are "Cauchy surfaces")

Ah I was not aware fo that
is it possible for it to be partially foliatable?

@JohnRennie: In the simple circuit, I discussed above, the following are the questions, point by point.

1. In the circuit I discussed above, would the electric field inside the ideal wires be zero?

2. If the electric field inside the ideal wires is zero then what is the cause behind it? Is it due to the charge accumulation across the resistance, such that it cancels the electric field completely due to the external battery?

3. Why the charge accumulates across the resistance? I have following hypothesis. Kindly comment.

I'm just thinking now about the curavture singularity in the schwarzschild metric, and how that might mess up being able to make time slices in some instances. However I'm pretty sure we can foliate schwarzschild spacetime in some places to get the spheres to define the schwarzschild coordinate

2:52 PM
@Charlie No, once it has a single Cauchy surface $S$, the manifold splits as $S\times \mathbb{R}$ with $\mathbb{R}$ being a timelike direction

ah I see a cauchy surface is just one such submanifold not the collection

@Charlie The singularity is simply not part of the manifold, i.e. you just have $\mathbb{R}^3 - \{0\}$ as the spatial slice of (non-extended) Schwarzschild

Ah ok
I'll have a think while I make my egg, thank you both

hey guys, I don't understand why div(v) = 0, would anyone be kind enough to explain this to me?

3:16 PM
@DevanshMittal hi, are you still around?

@JohnRennie: Yes Sir.
@JohnRennie: Sir, Kindly help me in the questions I posted above.
Do you want me to post them again for your reference?
Suppose we have a simple circuit with an ideal battery connected to resistance with ideal wires. We know that the potential difference across the resistance will be equal to "I*R", where I is the current and R is the resistance.
@JohnRennie: In the simple circuit, I discussed above, the following are the questions, point by point.

1. In the circuit I discussed above, would the electric field inside the ideal wires be zero?

2. If the electric field inside the ideal wires is zero then what is the cause behind it? Is it due to the charge accumulation across the resistance, such that it cancels the electric field completely due to the external battery?

3. Why the charge accumulates across the resistance? I have following hypothesis. Kindly comment.

No, it's OK.

I have posted them above again, in a new and better sequence.

What happens is that a battery produces free electrons at the anode and it consumes free electrons at the cathode. So if the battery is not connected there is a higher electron density in the metal at the anode than in the metal at the cathode i.e. there are more electrons per unit volume.

Yes Sir.
1. In the circuit I discussed above, would the electric field inside the ideal wires be zero?

3:32 PM
When the battery is connected to a resistor the excess electrons at the anode flow through the resistor to the cathode, but because the resistor resists the flow the electron density in the (ideal) wire between the anode and the resistor is higher than the electron density in the (ideal) wire between the cathode and the resistor.
This is a real physical difference, the electron densities really are different, but the difference is very small because the free electrons in the wires behave like a highly incompressible fluid.
That is, a small compression produces a very high potential difference.
Now for your question (1):

Wonderful explanation, Sir.
3. Why the charge accumulates across the resistance? I have following hypothesis. Kindly comment.
The electrons in the ideal wires are moving with some constant speed as the electric field there is zero. When they encounter the resistance, they are slowed down due to continuous collisions. Now there is a difference in speeds in the ideal wires and inside the resistance, due to which the charges accumulate across the resistance.

So, this explanation is probably correct?

There is a difference in the (average) speed of flow, but it is a very, very small difference. Too small to be measured.

OK Sir.
1. In the circuit I discussed above, would the electric field inside the ideal wires be zero?

The difference exists because the density of the free electrons on the anode side is fractionally greater than the density of the free electrons on the cathode side.

Yes Sir.
@JohnRennie And this difference comes because resistance behaves like a bottleneck in traffic?

3:38 PM
At a constant current the field in the (ideal) wires will be zero. That's because since they offer no resistance to the electron flow the density of the electrons is the same everywhere in the wire.
@DevanshMittal yes.

OHK Sir. Thanks a lot.

You can think of it as a fluid like water flowing through pipes. This is a surprisingly good analogy for electron flow.

Yes Sir.

Then the battery would be represented by a water pump, and the resistance by some obstruction to the flow.

2. If the electric field inside the ideal wires is zero then what is the cause behind it? Is it due to the charge accumulation across the resistance, such that it cancels the electric field completely due to the external battery?
@JohnRennie Yes Sir. I understand this analogy. But I also wished to understand exact what happens inside the wires, so I was not taking help of the analogy.

3:40 PM
@DevanshMittal the anode is constantly pumping electrons out into the (ideal) wire that is connected to it.

2. If the electric field inside the ideal wires is zero then what is the cause behind it? Is it due to the charge accumulation across the resistance, such that it cancels the electric field completely due to the external battery?

Since the wire has no resistance to the flow the electrons being pushed into it by the battery push other electrons out of the other end of the wire.
The electron density remains constant everywhere inside the wire, so there is no field within it.

Sir, Electron density being constant is the cause of electric field being zero, or electric field being zero is the cause of electron density being constant?

They are the same thing.

But what is more basic?

3:43 PM
If the electron density varied within the wire then electrons would want to flow from the high density to the lower density regions because of the electric field the density variations cause.

I believe, electric field being zero is the cause. Kindly comment.
And what is the cause of electric field being zero?

But if the wire has no resistance this flow would happen infinitely fast. i.e. any density variations that existed would cancel each other out in zero time.
Density variations cause a field, and the field causes a flow that cancels out the density variations. The two are linked.

Ohk Sir. Nice.
Sir, can we say that electric field inside the wire becomes zero because across the resistance the charge accumulation happens in such a way that the electric field between the anode and one end of resistance becomes zero?
And similar thing happens between cathode and other end of the resistance?

Yes

Ohk Sir. Thanks a lot.
Sir, Now let me come to the primary question which I have in my mind due to which I did all this discussion.

3:48 PM
OK ...

https://physics.stackexchange.com/questions/557743/what-is-the-meaning-of-potential-difference-in-presence-of-non-conservative-indu/

This is the question from where it all started.
I posted this question a few days ago and I still do not have a few answers.
I will take the same approach as earlier. I will split my original question into several basic chunks and move forward.
If you have read the question then following are my doubts.
1. Does the concept of potential difference really exist in the presence of time varying magnetic field or we just forcefully introduce it, even if it is conceptually wrong?

I'd prefer to take a step back and challenge your statement about the conservative nature of a circuit.

Yes Sir. Kindly correct my misconceptions.

If an electrical circuit was conservative we could take an electron right round the electron and the energy of the electron would not change.

OK Sir. I agree with that. I understand that inside the battery Non-Conservative forces are existing.
Sir, Is it true that the idea of potential and potential difference exists only for conservative force fields?

3:57 PM
OK. So you're happy that energy flows from the chemical reaction inside the battery, through the motion of the electron and out of the circuit as heat or light or whatever the circuit does? Yes?

Yes Sir.

In your example of the circuit in the magnetic field energy flows from the magnetic field, through the motion of the electrons and then out of the circuit as heat in the resistors.

Battery gives an electron the kinetic energy and some portion of that kinetic energy is lost in the collisions inside the resistance in the form of heat.

The behaviour is exactly the same as with a battery except that energy flows into the circuit from a magnetic field instead of from a chemical reaction.

WOnderful Explanation Sir.
So, can we say that the induced Non-Conservative electric field which get induced in the circuit due to changing magnetic fields is an agent of changing magnetic field and its task is like a mediator that it converts the magnetic energy into electrical energy and vice versa?

4:02 PM
Yes, that's a reasonable way of looking at it.

Thanks a lot, Sir.
Sir, Is it true that the idea of potential and potential difference exists only for conservative force fields? And the idea of Potential does not exist for non-conservative electric fields?

It's true, though there is a form of the potential that can be used despite the non-conservative nature of the fields. This is the magnetic vector potential.
The combination of the electric potential and magnetic vector potential work for all magnetic and electric fields.

Yes Sir. I read that somewhere, but I think that is well beyond the scope of my study. I have not read anything about magnetic vector potential.

But I have to confess that I don't know a lot about the vector potential as I had got bored with my electrodynamics course by that point :-)

Ohk Sir :)

4:07 PM
@Shing Because coordinates and velocities are independent in classical mechanics. see e.g physics.stackexchange.com/q/885/50583

My question is, in the kind of circuit I have discussed in my question above, how do we define the idea of potential difference between any two points when we do not see any existence of conservative electric field?
@JohnRennie

The potential difference between two points in the loop is, as always, just the work done per unit charge moved between those two points. The only problem comes if you go round the loop more than once when you move between the points.

Yes Sir, since in case of non-conservative electric fields, the potential difference between two points can have infinite values, depending on how many times we move in the loop, so is the question of calculation of potential different even valid?

@DevanshMittal yes, but all we have to do is define the potential as the work done from the start point to the first time we go through the end point.
You can't go round the circuit many times without going through the end point more than once.
That's because a circuit is effectively one dimensional i.e. electrons can only move along the wire between the two points.

@ACuriousMind oh!!!! I totally forgot about this. Thank you so much!

4:13 PM
This is very different to e.g. a magnetic field in 3D space where a charge has an infinite number of different paths from the start to end point.

Ohk Sir.

So, are we saying that the idea of potential difference here in presence of non-conservative electric field is only work done per unit charge?

And it is unrelated (or not completely related) to the idea of potential difference we define in presence of conservative electric field, where the potential of a point is fixed, inpdendent of the path?

So if you move from the start point to the end point then stop - don't pass through it and go round again - the potential difference is single valued.

Yes Sir.
Ohk Sir.

So, are we saying that the idea of potential difference here in presence of non-conservative electric field is only work done per unit charge?

And it is unrelated (or not completely related) to the idea of potential difference we define in presence of conservative electric field, where the potential of a point is fixed, inpdendent of the path?

Potential difference between two points is always the work done when a unit charge is moved between those points.

Ohk Sir.

So, can we say that it is wrong for us to look for conservative electric field always, when we are asked to calculated the potential difference?

4:17 PM
It's just that in a non-conservative field the work depends on the path taken between those points.

Ohk Sir. Thanks a lot. That's a big relief.
Can we say that in the circuit like above, there will be no existence of conservative electric field?
Wouldn't the charge accumulation here will happen the same way as it happened in a simple circuit containing battery and resistance and due to that charge accumulation, the electric field will also come into existence?

I think you're getting hung up on this issue.
The circuit is only non-conservative if you ignore the energy being pumped in by the magnetic field.

Ohk Sir.
Wouldn't the charge accumulation here will happen the same way as it happened in a simple circuit containing battery and resistance and due to that charge accumulation, the electric field will also come into existence?

That's like saying a regular circuit is non-conservative because you're ignoring the energy being pumped in by the battery.

Sir, The energy explanation I understood. Thanks to you.
I mean, wouldn't the conservative electric field also come into existence in the above circuit due to charge accumulation across the resistances?
Along with non-conservative electric field?
And these two will cancel in the ideal wires to make net electric field there zero? The same they cancelled in simple battery resistance circuit?

4:27 PM
The EMF from the magnetic field is different from a battery because it's like having an infinite number of tiny batteries distributed along the wire.

Yes Sir. I completely agree to that and I fortunately know that.
If the three branches have 3 resistances connected with ideal wires, then inside the ideal wires the net electric field should be zero. Is it correct, Sir?

hello

@DevanshMittal what's going to happen with the ideal wires is that the charge density changes along the wire. The charge density gradient creates a field that is equal and opposite to the field created by the changing magnetic field.

@JohnRennie: Yes Sir. I was asking exactly that.

@StanShunpike Hi :) What's up?

4:32 PM
@DevanshMittal in fact, I would need to sit down and think about this to be absolutely sure, but I think the charge density would change along the ideal wires but be constant inside the resistors.

@JohnRennie As soon as the current reaches a steady value, the electric field inside the ideal wires will become zero, as the conservative electric field developed due to charge accumulation across the resistances will completely cancel the non-conservative electric field there, developed due to changing magnetic fields.

Is it correct, Sir?

So it's exactly the opposite of what happens in a circuit with a single battery.

@ACuriousMind So yesterday I was watching this video on the spine. And I'm a bit confused how twisting a plane works in terms of storing potential energy
0

I agree with the commenter above that Why do we bend a book to keep it straight? has a very good explanation with detailed background. The short version is that when you twist a straight ruler, it bends but does not stretch. However, in order to twist a curved ruler, you need to stretch or compre...

here's the details
The part I'm confused about is why adding waves to a plane changes how the potential energy disipates when you twist the plane

@JohnRennie As soon as the current reaches a steady value, the electric field inside the ideal wires will become zero, as the conservative electric field developed due to charge accumulation across the resistances will completely cancel the non-conservative electric field there, developed due to changing magnetic fields.

Is it correct, Sir?

@DevanshMittal yes, or at least I am fairly sure that is the case.

4:34 PM
@JohnRennie Thanks a lot for the confirmation, Sir.

@DevanshMittal You don't need to keep repeating posts ...

@ACuriousMind the answer that was suggested as a duplicate talks about curvature. Which I'm sure is useful, but I also feel like none of the answers discussed talked about nodes

@JohnRennie Sorry Sir, I thought you missed that point, so I repeated.

and i'm just a little confused about the basic idea of why adding waves to a plane would change the way energy dissipates at all

If I take a while to answer it is because either I'm thinking about the question or I am doing something else. In neither case will repeating the question make me answer faster.

4:36 PM
@JohnRennie: So, now can we say that even in such a circuit with changing magnetic fields, we can define our old idea of potential difference which we define on the basis of conservative electric fields? As we have conservative electric field also in our circuit now?
@JohnRennie Sure Sir. I will remember that. Thanks and sorry again.

I think there is no problem defining a potential within the wire provided you take the path between the two points and don't allow multiple trips round the loop.

Yes Sir. The whole problem comes because when we teach electrostatics, we emphasis on the idea that potential is defined only for the conservative force field, so much that when it comes to changing magnetic fields case, the idea of potential here seems quite contradictory, so to convince our sharp students, we should have some good justification.

This is why I am confirming all these things with you.

@DevanshMittal The problem with non-conservative fields is when there are multiple different paths between the start and end points. Every path results in a different value for the potential difference. Yes?

@StanShunpike I'm not sure why the "node" would be relevant. Isn't the "why curving paper makes it rigid" part of physics.stackexchange.com/a/473105/50583 the answer to the question - when you've bent the ruler before twisting, you've "used up" the ability to curve extrinsically without curving intrinsically just like with the paper

@JohnRennie Absolutely Sir. Fortunately, our textbooks have not included questions of such kinds till now.
If we encounter such problems, then how would we respond to them?

4:43 PM
But in a circuit the electrons are constrained to flow along the wire so there is only one possible path between the start and end point. That means there is only one possible value of the potential difference between those two points. OK so far?

I am thinking on it, Sir.
If there are multiple branches with different resistance in parallel, then the potential difference across the two points about which all the branches are in parallel, can that be different across different paths? In case of changing magnetic fields?

That's a good question, and I will have to think about it.

OK Sir. You are my only hope. In my circle people do not know answers to these questions.

@ACuriousMind Ok, I guess I'll go study it again. Maybe I'm getting confused somewhere. Thanks for the feedback :)

@StanShunpike when you twist a ruler you are applying predominantly shear forces.
The ruler is thin so there is very little extension as it twists. Any form of twisting of an object is mostly shear.

4:53 PM
@JohnRennie well, the model I'm considering is the human spine
The author in question made an argument with a toy
but he didn't discuss the mathematics at all
so i am loooking for analogous systems
i can show u a clip
hang on
@JohnRennie I found the right moment. if you watch from here, it's like 30sec - 1 min and he presents his argument

@JohnRennie: I got some answer. Kindly consider the following figure.
Here, the potential difference between point A and B will turn out to be same for all the paths.
The EMF induced in the 3 loops corresponding to 3 resistance will be same.
Kindly comment.

@DevanshMittal Hmm, is that true? I'm not sure that it is.
But I have run out of time and have to go. We'll have to pick this up tomorrow.

Sir, I am sending you the solution.
@JohnRennie: Kindly see if above solution is correct.
It shows the Potential Difference to be the same.
@JohnRennie: Thanks a lot for the guidance, Sir.

Would love to learn from you next time when you are online.

1 hour later…
6:08 PM
it started to rain. Nevertheless, it still feels so hot indoors.
on my way back, I didn't see stars and planets (Jupiter, etc.) I usually see.
It's cloudy and eventually it rains now.
the indoor temperature is indicated as 32 degrees Celsius.
it's like today Jupiter is in opposition to the sun.

7:01 PM
0

A ship is approaching a cliff of height 105m from sea level. A gun fitted on the ship can fire shots with a speed of 100ms^{-1}. Find the maximum distance from the foot of the cliff from where the gun can hit an object on the top of the cliff So this is the question, I write the kinematic equat...

Guys
Can I get my question reopened? I tried my best to ask it.....
if someone can give me the answer here/ tell me how to get it reopened, that'd be awesome