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andrea96

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Thanks!

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In summary: B(t)}{dt} , which is the same as the electric field on a wire segment that is in contact with the changing magnetic field. The wire segment experiences a force due to the changing magnetic field, and that force is E \cdot l. The direction of the force is determined by the vector drawn from the center of the magnetic field to the point where the wire segment is located.

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andrea96

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Thanks!

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Chandra Prayaga

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It is not clear why you think the result should be independent of r. The flux does depend on the area of the circle, which depends on r.andrea96 said:

Thanks!

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andrea96

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rumborak

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willem2

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Dale

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The electric field does not depend on r, but the induced EMF does. Those are two different things. The EMF is the integral of E around the circle, which scales as r, just like you found.andrea96 said:but the electric field cannot depend on r.

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rumborak

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I see what willem and andrea are saying though, and it is a very interesting question. Why/how how does the wire segment know what direction the resulting electric field should point to? The wire segment is unaware of the rest of the circle after all.

My best guess is that it's a "net force" type of effect, where maybe neighboring line segments influence it.

I feel one should be able to construct Faraday's Law from the ground up through the Lorentz force. But that one already is only dependent on B, not the change of B, so i don't see how to construct the law. Apparently this is something that Einstein pondered and got him towards Special Relativity, so we're not pondering something stupid here.

EDIT: I think willem's suggestion of "there can not be a completely uniform field" might be the answer though. Magnetic fields are circular, so the field lines can not possibly be exactly parallel. When the field changes in time, different parts of the wire segment will experience different fields. The electrons in the wire will thus experience different Lorentz forces, which then likely results in the net physical force (and resulting electrical force) along the wire segment.

My best guess is that it's a "net force" type of effect, where maybe neighboring line segments influence it.

I feel one should be able to construct Faraday's Law from the ground up through the Lorentz force. But that one already is only dependent on B, not the change of B, so i don't see how to construct the law. Apparently this is something that Einstein pondered and got him towards Special Relativity, so we're not pondering something stupid here.

EDIT: I think willem's suggestion of "there can not be a completely uniform field" might be the answer though. Magnetic fields are circular, so the field lines can not possibly be exactly parallel. When the field changes in time, different parts of the wire segment will experience different fields. The electrons in the wire will thus experience different Lorentz forces, which then likely results in the net physical force (and resulting electrical force) along the wire segment.

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stedwards

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andrea96 said:

I'm going to back up a little. [itex]\oint E \cdot l = -\frac{d}{dt} \int\int B \cdot dA[/itex] .

For your circular path, [itex]2\pi r E = -\frac{d}{dt}2\pi r^2 B[/itex] .

You have a quantity you called circulation. This the the induced voltage around the loop. The EMF

I'll use the symbol V. [itex]V=2 \pi r E[/itex]

or

[itex]V=\frac{d}{dt}2\pi r^{2} B[/itex]

[itex]V=\frac{d\Phi}{dt}[/itex]

[itex]V=\frac{d\Phi}{dt}[/itex]

The voltage increases with [itex]r^2[/itex] because the quantity of Magnetic flux in the loop also increases with r.

If the magnetic flux were independent of r, and all located at near r=0, the induced voltage would be independent of the path. This may be the invariant you were thinking of.

This why the voltage out of a power transformer depends on the number of secondary turns, but not their shape and size to any great degree, because most of the magnetic flux is in the iron core.

Does this help? I could be addressing the wrong issue.

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andresB

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willem2 said:But what is the direction on the electric field? any point lies on many different circles with tangents in all possible direction. Symmetry seems to require that the electric field is zero everywhere..

I have the same feeling.

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willem2

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[tex] E = (r - r_0) × \frac {\delta B}{\delta t} [/tex] is solution for every r_0,

You can also combine two or more solutions, if E1 and E2 are solutions, so is [itex] c_1 E_1 + c_2 E_2 [/itex] as long as [itex] c_1+c_2 = 1 [/itex]

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andrea96

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Perfect! This is the paradox: I can obtain infinity different results for each direction of the electric field. You have said that the explanation of this paradox is the impossibility to creat that magnetic field: this could be an interesting answer, but I'd like to inivite you to think that it's impossible to build also an infinity plane or linear distribution of charge, but in these case the electric field is well definite [tex] E=\frac{\sigma}{2\epsilon} [/tex] or [tex] E=\frac{\lambda}{2\pi \epsilon r} [/tex]. I'd like to show you also that a similar paradox came out if I want calculate the electric field in an infinity and uniform 3-dimentional distribution of charge: the field should be 0 since the simmetry, but calculating it using an arbitray Gaussian surface, I can obtain any value of the electric field chooising any Gaussian surface. And you can see as other two similar paradoxes exist: infinity 3-dimentional current distribution and infinity uniforno-costant electric field. There is a common reason in all these four paradoxes. Instead, this "problem" there isn't in the cases of infinity uniform plan and linear distribution of charge...willem2 said:

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Dale

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I don't know how your confusing EMF around a loop with the E field at a point constitutes a paradox.

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andrea96

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Maybe I have explained vary bad: I used the EMF to calculate the E field, the result about EMF isn't a paradox (it obviously depend on r), but the result of E filed constitutes a paradox. I try to explain better: in the space there is a uniform infinitely extended B field that is changing in time. This variation of magnetic field due an E field. To calculate the electric field in a certain point I calculate the electric circuitation on a circle (the circle must pass on the point in which I want to calculate the E filed) and exploiting the symmetry of the system ( from the infinity extention of the B field, the inducted E field has to be the same in any point on the circle ) I can calculate the E field : [tex] E=\frac {EMF}{2\pi r}=\frac {-\pi r^2\frac {dB (t)}{dt}}{2\pi r}=-\frac {r}{2} \frac {dB (t)}{dt} [/tex].DaleSpam said:I don't know how your confusing EMF around a loop with the E field at a point constitutes a paradox.

From the result is evident that the E field depend on the particular choise of the circle. Now choosing an other circle passing in the point where I'm calculating the electric field with an other radius r', I obtain a different result for the E field : [tex] E=-\frac {r`}{2} \frac {dB (t)}{dt} [/tex]. And this is a paradox: I have obtained two different values for the E field in a certain point, but the field should be univocally determinated...

I hope that my leatest explanation was better...

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rumborak

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If I understand his andrea's point correctly, and it's what I was getting to also, is how can the local field be dependent on the rest of the circle?

And yes, I know of course that the Maxwell/Farady equation say exactly that. But it still begs the question how this "holistic" result can arise when in the end, it's just singular electrons in a wire experiencing their local magnetic field.

And yes, I know of course that the Maxwell/Farady equation say exactly that. But it still begs the question how this "holistic" result can arise when in the end, it's just singular electrons in a wire experiencing their local magnetic field.

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- #15

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$$\vec{\nabla} \times \vec{E}=-\frac{1}{c} \dot{\vec{B}}(t).$$

That's a constant wrt. to spatial variables. So we look for a solution of this seen as a differential equation. The solution is determined only up to a gradient field

$$\vec{E}=\frac{1}{2c} \vec{x} \times \dot{\vec{B}}(t)-\vec{\nabla} \Phi(t,\vec{x}).$$

So ##\vec{E}## won't be homogeneous. Of course, you need more information to calculate the electric field completely.

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nasu

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This (second line, left hand side) will be true only if the electric field will the tangent to that circle in all points of the circle and have the same magnitude at all points.stedwards said:I'm going to back up a little. [itex]\oint E \cdot l = -\frac{d}{dt} \int\int B \cdot dA[/itex] .

For your circular path, [itex]2\pi r E = -\frac{d}{dt}2\pi r^2 B[/itex] .

So you are assuming already some geometry of the field.

More general, the integral around the circle will give you info just about the tangential component of the field.

You cannot find the entire field from that unless the field happens to be tangential to some circle.

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andresB

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rot B= dE/dt, but

rot B =0, since the magnetic field is uniform and only depend on time. so the electric field doesn't depend on time.

But, on the other hand

Rot E= - dB/dt

and the right hand side can depend on time since B(t) is arbitrary, but then rot E will depend on time in contradiction with the previous result.

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Dale

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This calculation is incorrect. See Vanhees post for the correct E field. You can plug it back in and see that it satisfies Maxwell's equations, whereas yours does not. However, ...andrea96 said:I can calculate the E field : [tex] E=\frac {EMF}{2\pi r}=\frac {-\pi r^2\frac {dB (t)}{dt}}{2\pi r}=-\frac {r}{2} \frac {dB (t)}{dt} [/tex].

Even if your expression were correct, it would still not constitute a paradox. You have simply incompletely specified the boundary conditions so you have not determined a unique solution to Maxwell's equations. Not only is there a valid solution to Maxwell's equations which satisfies your conditions, there are an infinite number of such solutions. Having multiple solutions does not constitute a paradox.andrea96 said:And this is a paradox: I have obtained two different values for the E field in a certain point, but the field should be univocally determinated..

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andrea96

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Thanks for your answer! However my expression for E doesn't satisfy Maxwell equation because, as I said in one of my initials post, it isn't the vectorial electric field, but just his component along the tangential direction. However I understand what you are saying about the unicity of the solution, but if I specify that the magnetic field is generated by a very big coil? What happens?DaleSpam said:This calculation is incorrect. See Vanhees post for the correct E field. You can plug it back in and see that it satisfies Maxwell's equations, whereas yours does not. However, ...

Even if your expression were correct, it would still not constitute a paradox. You have simply incompletely specified the boundary conditions so you have not determined a unique solution to Maxwell's equations. Not only is there a valid solution to Maxwell's equations which satisfies your conditions, there are an infinite number of such solutions. Having multiple solutions does not constitute a paradox.

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Dale

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I think that is correct then for lops centered on the origin and 0 gradient field. It may be that there is some relationship between the center of your loop and the gradient.andrea96 said:Thanks for your answer! However my expression for E doesn't satisfy Maxwell equation because, as I said in one of my initials post, it isn't the vectorial electric field, but just his component along the tangential direction.

Assuming that you also specify that there are no other sources and some appropriate boundary conditions (absorbing, reflecting, zero at infinity, or whatever) then you could find a unique solution. In the center of the coil it should look like Vanhees' solution with 0 gradient field in coordinates with the origin at the center of the loop.andrea96 said:However I understand what you are saying about the unicity of the solution, but if I specify that the magnetic field is generated by a very big coil? What happens?

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andrea96

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Thank you very much!

An apparent paradox in no constant fields refers to a situation where two seemingly contradictory or illogical outcomes occur in the absence of any constant or external factors. This can be observed in scientific experiments or theories that do not follow traditional laws or principles.

An apparent paradox in no constant fields challenges traditional scientific principles by demonstrating that there may be exceptions or limitations to these principles. It forces scientists to question their existing understanding and theories, and to explore new possibilities.

One example of an apparent paradox in no constant fields is the double-slit experiment in quantum mechanics, where particles can behave as both waves and particles at the same time. Another example is the theory of relativity, which challenges the notion of a constant and absolute time and space.

Scientists resolve apparent paradoxes in no constant fields by conducting further research and experiments to gather more data and evidence. They also revise or develop new theories and principles to explain these phenomena.

Apparent paradoxes in no constant fields can have a significant impact on scientific progress as they can lead to the discovery of new phenomena, theories, and principles. They can also challenge traditional ways of thinking and open up new areas of research and exploration.

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