Understanding Induced Electric Fields in a Uniform Magnetic Field

[LucasGB]

Suppose I have a perfectly constant, uniform magnetic field extending over space. Now I imagine a circle in this space. Now I imagine the circle is growing larger. The magnetic flux through this imaginary circle is, therefore, increasing. Therefore, there must be an induced electric field. But how can I have created an electric field by imagining things?

 

[Born2bwire]

I guess it could work. What you would do in reality is have a long wire. You would loop this wire into your loop in the plane normal to your magnetic fields and then run the wire through a slip knot so that the leftover wire is hanging down along the magnetic field vectors. This way, you could pull on the wire to open and close the loop, the excess wire being pulled down along the field vectors.

In this way you would be providing a mechanical input into the system. By moving the wire, you are accelerating the charges in the wire since they are confined by the wire. It seems to be conceptually the same as the classic case of when we have a loop of wire entering and exiting a static magnetic field.

 

[netheril96]

Suppose I have a perfectly constant, uniform magnetic field extending over space. Now I imagine a circle in this space. Now I imagine the circle is growing larger. The magnetic flux through this imaginary circle is, therefore, increasing. Therefore, there must be an induced electric field. But how can I have created an electric field by imagining things?

There is NO electric field here.The EMF is due to the motion of the wire,or Lorentz force in micrsoscopic scale,not a newly induced e-field.

 

[Bob S]

What you have described is very close to a particle accelerator called a betatron. Instead of an imaginary circle growing larger in a constant magnetic field, the betatron has a constant diameter toroidal vacuum chamber (your imaginary circle), and a varying magnetic field inside the toroid. The volts per turn (at 60 Hz) is given by Faraday's Law:

V(t) = -(dB/dt)∫B(t)·dA = 377·A·B_max·sin(ωt)

The largest of these accelerators, built at the University of Chicago in 1949, had a ~1.5-m diameter toroidal vacuum chamber, and roughly 1 square meters of transformer iron that ran off the ac line frequency, such that the magnetic field cycled from ~- 1.4 Tesla to + 1.4 Tesla at 60 Hz. The peak azimuthal electric field inside the vacuum chamber was roughly 200 volts per meter. Electrons from a hot filament were accelerated in this azimuthal field inside the vacuum chamber to over 300 million volts during the half-cycle where dB/dt was the right polarity. See photo of betatron

http://storage.lib.uchicago.edu/apf/apf2/images/derivatives/apf2-00056r.jpg

See theory

http://teachers.web.cern.ch/teachers/archiv/HST2001/accelerators/teachers notes/betatron.htm

Bob S

 

[LucasGB]

Thank you all for your replies.

There is NO electric field here.The EMF is due to the motion of the wire,or Lorentz force in micrsoscopic scale,not a newly induced e-field.

Yes, that seems very reasonable. But Faraday's Law of Induction clearly states that the circulation of the electric field around a curve equals minus the rate of change of the flux through a surface defined by that curve with time. The flux through my imaginary circle is changing. Where's the electric field?

 

[netheril96]

Thank you all for your replies.



Yes, that seems very reasonable. But Faraday's Law of Induction clearly states that the circulation of the electric field around a curve equals minus the rate of change of the flux through a surface defined by that curve with time. The flux through my imaginary circle is changing. Where's the electric field?

You got it wrong.What Faraday's Law of Induction states is the electromotive force equals minus the rate of change of the flux through a surface

 

[LucasGB]

You got it wrong. What Faraday's Law of Induction states is the electromotive force equals minus the rate of change of the flux through a surface

So there's two Faraday's Laws? One for when the conductor moves and one for when the conductor stands still? Because I'm pretty sure the third Maxwell Equation is what I described.

 

[netheril96]

So there's two Faraday's Laws? One for when the conductor moves and one for when the conductor stands still? Because I'm pretty sure the third Maxwell Equation is what I described.

Show your formula

 

[LucasGB]

Show your formula

Check the attachment.

 

[netheril96]

Check the attachment.

Please type LaTex next time

The formula you give is Maxwell equations,not Faraday's law of induction

 

[LucasGB]

Please type LaTex next time

The formula you give is Maxwell equations,not Faraday's law of induction

I apologize, but I don't know how to use LaTeX very well. The formula I gave you is both Faraday's Law of Induction and one of Maxwell's Equations.

All 4 Maxwell's Equations have alternative names. Respectively: Gauss's Law for Electric Fields, Gauss's Law for Magnetic Fields, Faraday's law of Induction and Ampère's Circuital Law.

 

[bjacoby]

Thank you all for your replies.

Yes, that seems very reasonable. But Faraday's Law of Induction clearly states that the circulation of the electric field around a curve equals minus the rate of change of the flux through a surface defined by that curve with time. The flux through my imaginary circle is changing. Where's the electric field?

It's obvious the IMAGINARY electric field has a circulation about your imaginary surface!

You are of course being cute, but the question in a large sense can be quite serious. We could word it this way: Are thoughts things? Since mathematics is abstract, we could ask "Is mathematics more real than reality?" The questions are real though physics at present is EXTREMELY reluctant to tackle them.

As a general introduction let's just note a couple of things about modern physics and these issues. First, one might generate a theory that the physical universe has more orthogonal dimensions than three or four. One could imagine that thoughts exist in parallel dimensions in other similar but orthogonal three dimensional spaces. Thought, for example, might exist in such a space that we might term, Oh, I don't know, maybe we could call it the "astral space" or "etheric space". Many physicists vehemently deny this possibility. And yet they then turn around and come up with string theory that proposes maybe 11 such dimensions. And then they turn around and deny it again strongly asserting that reality only has THREE dimensions but then come up with a cosmology that requires four dimensions so that the universe can be everywhere expanding!

Obviously there is a lot of confusion and conflicting thoughts here. And all that is made worse by a dogmatic attitude that rejects a variety of subjects such as extra dimensions or the reality of thought beyond mere brain functioning. Quite obviously physics is quite primitive at present and many people intend to keep it that way.

 

[netheril96]

I apologize, but I don't know how to use LaTeX very well. The formula I gave you is both Faraday's Law of Induction and one of Maxwell's Equations.

All 4 Maxwell's Equations have alternative names. Respectively: Gauss's Law for Electric Fields, Gauss's Law for Magnetic Fields, Faraday's law of Induction and Ampère's Circuital Law.

I use MathType to spare the pains of learning LaTex's grammar.

If I don't get it wrong,\[\oint_{\partial S} {\vec E \cdot d\vec l = - \frac{{\partial \Psi }}{{\partial t}}} <br /> \]is the Faraday's induction law
BUT you should notice that the E here is NOT electrostatic field strength.Instead,it is non-electrostatic strength like Lorentz force or vortex electric field strengh.And here,it is Lorentz force rather than vortex e-field

 

[LucasGB]

It's obvious the IMAGINARY electric field has a circulation about your imaginary surface!

You are of course being cute, but the question in a large sense can be quite serious. We could word it this way: Are thoughts things? Since mathematics is abstract, we could ask "Is mathematics more real than reality?" The questions are real though physics at present is EXTREMELY reluctant to tackle them.

As a general introduction let's just note a couple of things about modern physics and these issues. First, one might generate a theory that the physical universe has more orthogonal dimensions than three or four. One could imagine that thoughts exist in parallel dimensions in other similar but orthogonal three dimensional spaces. Thought, for example, might exist in such a space that we might term, Oh, I don't know, maybe we could call it the "astral space" or "etheric space". Many physicists vehemently deny this possibility. And yet they then turn around and come up with string theory that proposes maybe 11 such dimensions. And then they turn around and deny it again strongly asserting that reality only has THREE dimensions but then come up with a cosmology that requires four dimensions so that the universe can be everywhere expanding!

Obviously there is a lot of confusion and conflicting thoughts here. And all that is made worse by a dogmatic attitude that rejects a variety of subjects such as extra dimensions or the reality of thought beyond mere brain functioning. Quite obviously physics is quite primitive at present and many people intend to keep it that way.

Whoa. Look, the question is not metaphysical or philosophical. I'm really making a mistake in understanding the physics, and I'm asking people to help me see where the mistake is. But I liked what you wrote anyway.

 

[LucasGB]

I use MathType to spare the pains of learning LaTex's grammar.

If I don't get it wrong,\[\oint_{\partial S} {\vec E \cdot d\vec l = - \frac{{\partial \Psi }}{{\partial t}}} <br /> \]is the Faraday's induction law
BUT you should notice that the E here is NOT electrostatic field strength.Instead,it is non-electrostatic strength like Lorentz force or vortex electric field strengh.And here,it is Lorentz force rather than vortex e-field

Thank you for the MathType tip, I'll look into that.

Look, in that formula, E really isn't "electrostatic field strength". It represents the electric field vector, an induced electric field, as opposed to electrostatic. Nevertheless, it is an electric field. Therefore, if I imagine that the magnetic flux through the disk of my imaginary circle is changing because the circle is increasing, then the equation you just used says there must be an induced electric field circulating around the circle. That's what the E.dl integral means. Now, this can't be right, for I can't create an electric field by imagining things, and define its intensity by imagining how fast the imaginary circle changes size. So there's a mistake in my analysis. I don't know where it is.

 

[netheril96]

Thank you for the MathType tip, I'll look into that.

Look, in that formula, E really isn't "electrostatic field strength". It represents the electric field vector, an induced electric field, as opposed to electrostatic. Nevertheless, it is an electric field. Therefore, if I imagine that the magnetic flux through the disk of my imaginary circle is changing because the circle is increasing, then the equation you just used says there must be an induced electric field circulating around the circle. That's what the E.dl integral means. Now, this can't be right, for I can't create an electric field by imagining things, and define its intensity by imagining how fast the imaginary circle changes size. So there's a mistake in my analysis. I don't know where it is.

Go find some stuff about EMF and then discuss about this problem
You problem is your misunderstanding of what is an electromotive force

 

[LucasGB]

Go find some stuff about EMF and then discuss about this problem
You problem is your misunderstanding of what is an electromotive force

Let's wait until tomorrow and see what other people have to say about this. Thank you for your help, though.

 

[LucasGB]

So, any ideas on this?

 

[Bob S]

Faraday's Law for inducing an electric field E (NOT a current) in a loop of length L surrounding the area A of induction:

1) ∫E·dl = E·L = -(d/dt)∫B·n dA = -A·dB/dt

and

2) ∫E·dl = E·L = -(d/dt)∫B·n·dA = -B·dA/dt

The expanding virtual loop in the OP is case 2) with a constant magnetic field and increasing area A. This form of Faraday's Law applies to generators with rotating armatures in a constant stator field.
The constant area virtual loop with a changing enclosed magnetic field (case 1) represents the betatron particle accelerator in my earlier post #4. The induced azimuthal electric field is a closed (virtual) loop in vacuum, without wires, with or without free electrons.

Bob S

 

[LucasGB]

Faraday's Law for inducing an electric field E (NOT a current) in a loop of length L surrounding the area A of induction:

1) ∫E·dl = E·L = -(d/dt)∫B·n dA = -A·dB/dt

and

2) ∫E·dl = E·L = -(d/dt)∫B·n·dA = -B·dA/dt

The expanding virtual loop in the OP is case 2) with a constant magnetic field and increasing area A. This form of Faraday's Law applies to generators with rotating armatures in a constant stator field.
The constant area virtual loop with a changing enclosed magnetic field (case 1) represents the betatron particle accelerator in my earlier post #4. The induced azimuthal electric field is a closed (virtual) loop in vacuum, without wires, with or without free electrons.

Bob S

I see. So if there's a constant magnetic field and no electric field, and I merely imagine a size-changing circle, then according to 2) there must be an induced electric field? Isn't this problematic?

 

[Bob S]

I see. So if there's a constant magnetic field and no electric field, and I merely imagine a size-changing circle, then according to 2) there must be an induced electric field? Isn't this problematic?

Correct. If you have a size-changing circle in a constant magnetic field, there must be an induced electric field around the circle. No wire required.

Bob S

 

[LucasGB]

Alright, now we're getting somewhere.

But here's the catch: the faster I imagine the circle to be changing its size, the larger the rate of change of flux and the larger the induced electric field. So apparently, I can make the electric field as big as I want by changing how fast the circle is changing its size. But what is the size of the real electric field?

And what if I don't imagine any circle at all? Then there's no electric field?

 

[Bob S]

And what if I don't imagine any [expanding] circle at all? Then there's no electric field?

This is one of the puzzling examples of electromagnetism. There is certainly an electric field around the expanding circle, so there is an electric field between each point defining the imaginary circle. But if you select just two points, and ask if there is an electric field between them, the logical answer is no. Specifically, for the expanding circle of radius R in a constant magnetic field B (using Faraday's Law):

2πR·E = - B·dA/dt = - B·d(πR²)/dt = -2πR·B·dR/dt, or

E = -B·dR/dt

So there is an electric field everywhere on the imaginary expanding circle. So the motion (dR/dt) induces the electric field.

Bob S