Induced Electric Field Lines When a Magnetic Field Crosses a Plane

[aim1732]

If a time varying magnetic field of infinite dimensions crosses a plane at right angles what can we expect induced electric field lines to look like? Straight lines with direction determined by the right hand rule?

 

[zorro]

No.
The induced electric field forms a vortex (concentric circular loops).
Its direction is determined by right hand grip rule.

 

[aim1732]

And what will the center of such a vortex be?[The question was asked keeping this very fact in mind]

 

[zorro]

No, the question was asked thinking about electric field lines to be straight
The plane is infinite and so is the magnetic field. You can arbitrarily take any point as the centre.

 

[aim1732]

That does not uniquely fix the direction of field at any point----an essential condition for any vector field.

 

[zorro]

Offcourse it does. Fix a centre of the vortex - it will be the geometric centre if the plane is finite. But if its infinite - it can be assigned arbitrarily. After this step, the direction of field at any point is uniquely defined - tangential to the line joining the centre and the point.

 

[aim1732]

I don't understand------if your first step does not uniquely define the centre of vortex how do you expect the second step to? Consider this---the centre can lie anywhere and hence the field can have any direction at that point------yes it is wrong and makes no sense. I can not simply say any direction because the direction will always be unique no matter what and any contradiction of this fact is a definite indicator of flawed logic.

 

[zorro]

The problem here is that the plane is infinite. In this case it is same like fixing a co-ordinate system with an arbitrarily located origin and then doing vector algebra w.r.t. this origin which was arbitrarily fixed. Now will you say that the centre (origin) can lie anywhere in the plane and hence any vector under consideration can have any direction (w.r.t. the origin)?

If you have a better solution then let me know

 

[hikaru1221]

I'm not sure how the B-field looks like, but you may want to consider the picture

 

[zorro]

I'm not sure how the B-field looks like, but you may want to consider the picture

A plane is infinite. You cannot make it finite (keeping the closed loops around the plane in mind) . The B-field is just perpendicular to the plane everywhere (thats how I interpret the question).

 

[hikaru1221]

What I mean is that such field does not exist (in the sense that the field is truly infinite) or just a small small small portion of the real field (in the sense that we consider the field to a small extent)

 

[zorro]

Well it might be possible that such infinite field cannot exist in reality as you say. But we have to accept the question at its face value without arguing against it

You cannot consider a small part of the field and form closed loops when it is given that it is of infinite dimensions.

 

[hikaru1221]

But we have to accept the question at its face value without arguing against it

I'm not saying it doesn't exist in just real life. I'm saying it doesn't exist, even in theory
It's just like asking how your house would look like if you ran at v = 10c.

 

[stevenb]

I'm not saying it doesn't exist in just real life. I'm saying it doesn't exist, even in theory
It's just like asking how your house would look like if you ran at v = 10c.

Yes, and your mention of the speed of light is also relevant here because not only is there and issue with infinite spatial extent and the need for a real center, but there is an issue with propagation speed or retardation effects.

Good paradox question from the OP though.

 

[zorro]

I'm saying it doesn't exist, even in theory

Prove

This is hypothesis (the style in which the question is given - a magnetic field of infinite dimensions) and everything is possible in hypothesis. There are several such physics questions in which something is not possible - but you have to assume it and solve the problem

 

[stevenb]

Prove

Of course it is appropriate to say this should be proved, but I dont' think it would be too hard to show (assuming it's true) that the existence of an infinite uniform time rate of change of magnetic field is incompatible with Maxwell's equations.

Some hypothetical thought experiments (infinite plane waves, infinitely long wires, uniform charge density over all space etc.) are at least compatible with Maxwell's equations even if they are not physically realizable.

I think this is hikaru's assertion, and I'm inclined to agree. But, still I would reserve final judgement until I see a formal proof. I'll look at this over the weekend if someone does not post a proof (either proving or disproving) before then.

 

[zorro]

Some hypothetical thought experiments (infinite plane waves, infinitely long wires, uniform charge density over all space etc.) are at least compatible with Maxwell's equations even if they are not physically realizable.

I did not know that. Can you show how they are compatible?

 

[zorro]

Btw Hikaru I was just kidding. I don't want to waste your time to find its proof. I know you are a busy person

 

[hikaru1221]

I would like to leave the issue of the upper limit of propagation speed to stevenb I would not dare to touch wave equations in 3D; sorry for my shortcoming.

My assertion is based on the assumption that the OP was talking about an infinite B-field with a lack of a center. The B-field must come from some source, either a current setup or varying E-field. But even if it's varying E-field, the E-field must in turn come from some real source, which is charge setup (Abdul Quadeer, you may want to look back to my post in one of your thread ). The setup cannot be infinitely large, so a center should be determined (see my picture in my previous post). If the setup is infinitely large, the energy needed to drive this source is infinite.

@Abdul Quadeer: They are compatible if there is no violation Think of my example of v=10c, an obvious violation Another example is perfect conductor (not to confuse with superconductor) - the ideal wire we always have in our circuit diagram, or the ideal string and massless pulley; they are valid extension of the reality to the limit (resistivity of ideal wire = 0; tension in ideal string is uniform; mass of ideal pulley = 0).

@stevenb: If I use a plane wire carrying varying current, the B-field at every point in the plane of the wire should be perpendicular to the plane, regardless of retardation, I guess. If so, I still have an infinite varying B-field which crosses a plane at right angle and matches the criteria of the OP

 

[stevenb]

@stevenb: If I use a plane wire carrying varying current, the B-field at every point in the plane of the wire should be perpendicular to the plane, regardless of retardation, I guess. If so, I still have an infinite varying B-field which crosses a plane at right angle and matches the criteria of the OP

Yes, I see your point. I seem to have subconsciouly added more restrictions than stated originally. I was thinking in terms of uniform time changing field over the plane. I definitely need to think about this more. It's an interesting question.

 

[zorro]

My assertion is based on the assumption that the OP was talking about an infinite B-field with a lack of a center. The B-field must come from some source, either a current setup or varying E-field. But even if it's varying E-field, the E-field must in turn come from some real source, which is charge setup (Abdul Quadeer, you may want to look back to my post in one of your thread ). The setup cannot be infinitely large, so a center should be determined (see my picture in my previous post). If the setup is infinitely large, the energy needed to drive this source is infinite.

@Abdul Quadeer: They are compatible if there is no violation Think of my example of v=10c, an obvious violation

We have situations like this- an infinitely long current carrying wire. Where does the source of driving this infinite charge come from? Is it not a violation?

The OP's latest question was regarding the logic in fixing a centre. I rechecked your picture and found that you too arbitrarily fixed a centre in the plane.

 

[hikaru1221]

We have situations like this- an infinitely long current carrying wire. Where does the source of driving this infinite charge come from? Is it not a violation?

The term "infinitely long" straight wire in particular (and "very large setup" in general) is normally used in the meaning that we consider the phenomenon locally: we don't look at the field at the end of the wire. The wire cannot be infinitely long in the meaning that it is truly infinite. This is what I'm trying to say by the picture. If you are alone in a desert, things would look like to you that you would never get out of it; but you can if lucky enough, right? Similarly, if you look at the space locally, it would look like to you that the setup is infinite and the field is just the same at every other points in the space, but they are actually not.

The OP's latest question was regarding the logic in fixing a centre. I rechecked your picture and found that you too arbitrarily fixed a centre in the plane.

The emphasis of the picture is on the current source. This is where the center comes from. The imaginary plane has nothing to do with the center of the field. The shape of the field depends on the source, and thus, since the source is finite, that constraints the field in a sense that there is a center.

 

[zorro]

Similarly, if you look at the space locally, it would look like to you that the setup is infinite and the field is just the same at every other points in the space, but they are actually not.

How can you ascertain that the field is not same everywhere? It may or may not be depending upon the source.

The emphasis of the picture is on the current source. This is where the center comes from. The imaginary plane has nothing to do with the center of the field. The shape of the field depends on the source, and thus, since the source is finite, that constraints the field in a sense that there is a center.

How will you define a magnetic field of infinite dimensions (given in the question) from a finite source?

 

[hikaru1221]

How can you ascertain that the field is not same everywhere? It may or may not be depending upon the source.

Look at the picture. The B-field has closed field lines. At some point, B points upwards; but at some other points, B points downwards. It is the nature of the field.

By the way, you may want to prove that uniform infinite B-field doesn't exist in the constraint of finite source, say, a current source. Hint: Apply Ampere Law at the current.

How will you define a magnetic field of infinite dimensions (given in the question) from a finite source?

B-field is inherently infinite (in the extent of classical physics). It's just that it must have a determined center - that's the point.

 

[zorro]

By the way, you may want to prove that uniform infinite B-field doesn't exist in the constraint of finite source, say, a current source. Hint: Apply Ampere Law at the current.

The field is time-varying, not uniform.
'Apply Ampere Law' to what?

B-field is inherently infinite (in the extent of classical physics). It's just that it must have a determined center - that's the point.

I don't get you. In the first quote you say that infinite B-field doesnot exist for a finite source. In the second quote you say that it is infinite .

 

[hikaru1221]

The field is time-varying, not uniform.
'Apply Ampere Law' to what?

"Time-varying" is the opposite of "constant"; "uniform" is the opposite of "non-uniform"
Anyway I just mean to show you that a uniform infinite B-field doesn't exist. Don't bother that problem until you get through this

I don't get you. In the first quote you say that infinite B-field doesnot exist for a finite source. In the second quote you say that it is infinite .

No, what I mean is that infinite B-field without a center doesn't exist And uniform B-field doesn't exist either. B-field can be uniform locally, but when we look at the whole picture, it is not uniform.

P.S.: I think the issue of uniformity, infinity and center thingy is the source of confusion. People usually visualize an infinite field as a field with straight field lines only, i.e. uniform field. If such field exists, it has no center (so "uniform" = "no center"). However, fields of Nature are usually infinite, whereas they are actually non-uniform. "Infinite" is not "uniform".

Another source of confusion is that people usually use the words "infinite" and "uniform" without being aware of the constraints. They are NOT using those words literally. "Infinite" only means very large / spreading over a large space, and "uniform" only means approximately the same in the considered space.

 

[aim1732]

Good to see some interest.

The setup hikaru showed is just one of the possibilities----- I had an infinite wire carrying current in mind whose current varies with time. At distances very far away from the wire we can draw planes containing the current source such that the field crosses perpendicularly,is nearly straight(a result of large distance from source) and is very nearly constant. But it would still have an axis of symmetry by virtue of being produced by a current source---that is the wire itself.How do you suggest the vortex field lines to then look like? If the wire were finite we would have taken the centre of the wire as the centre of the vortex but the wire is infinite and it's field is quite different from that of a finite source.[If the center of vortex were to lie on the wire it would make the induced E field lines very nearly straight too as I said in my first post].

I know you would say an infinite wire can still have a centre--like it is divided into two rays infinite on either side. I am not going to argue about that anymore because I realize well that something not existing in nature can not be expected to be governed by physical laws.
An interesting related situation(rather simple): The E field of an infinite charged plane is independent of distance. It is useful when calculating stuff for it but ever tried to calculate a potential for it? It is strange but I think that in order to predict hypothetical situations sometimes we need to drop some of the more basic ideas,as in,what kind of conservative electrostatic field requires infinite work to b performed on a charge to just bring it in the frame?

 

[hikaru1221]

The setup hikaru showed is just one of the possibilities----- I had an infinite wire carrying current in mind whose current varies with time.

The setup can never be truly infinite, if it's a closed circuit First, you need to close the circuit. Second, the energy to drive the circuit cannot be infinite.
I can give you an example of a truly infinite current: you use some mechanism that "shoots" electrons to free space. The electrons expand in some directions in the space and may form infinite currents (currents that are not confined within limited space). However, the true source of this current is the mechanism, and this mechanism, again, cannot be infinite.

How do you suggest the vortex field lines to then look like? If the wire were finite we would have taken the centre of the wire as the centre of the vortex but the wire is infinite and it's field is quite different from that of a finite source.[If the center of vortex were to lie on the wire it would make the induced E field lines very nearly straight too as I said in my first post].

E-field lines are approximately straight at the points near the wire. If you think of a closed circuit at a larger extent, the vortex will be somewhere within the loop. The vortex is of the E-field as a whole and thus can be at anywhere in the space, so we cannot look at the picture locally at points near the wire - we have to look at the whole picture.

something not existing in nature can not be expected to be governed by physical laws.

As stevenb said, even if it is not realizable in reality, it is still valid as long as there is no violation with the law of nature. However, an infinite setup is not that case.

An interesting related situation(rather simple): The E field of an infinite charged plane is independent of distance. It is useful when calculating stuff for it but ever tried to calculate a potential for it? It is strange but I think that in order to predict hypothetical situations sometimes we need to drop some of the more basic ideas,as in,what kind of conservative electrostatic field requires infinite work to b performed on a charge to just bring it in the frame?

I think that's because a uniform E-field says much more than a field with V=Cx (C=const) No one forbids you to solve a free fall problem with Lagrangian.

 

[aim1732]

No one forbids you to solve a free fall problem with Lagrangian.

Sadly I do not know that much calculus as yet.

 

[hikaru1221]

Sadly I do not know that much calculus as yet.

Basically I mean, some people find it easier to work directly with E-field in that case, instead of its potential But convenience is subjective