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I have encountered a "paradox" while studying electromagnetism and I can't seem to understand where I am messing up.

Consider a uniform, time-dependent magnetic field B(t) = $t^4 \hat{i}$ in some region of space.

By Faraday's Law, there must be some induced electric field due to the changing magnetic field. However, since the magnetic field is uniform, any two points in the space are indistinguishable, so I believe that the electric field must also be uniform everywhere in that region of space. However, if the electric field at a point is independent of its position in space, then the E field must have curl 0.

So by Faraday's Law: $\nabla \times E = -dB/dt$. However, as per my above reasoning, the left side of this equation is 0 whereas the right side will be proportional to $t^3$? How can this be?

I have a few guesses as to where I went wrong, but no justification for why they should be true:

  1. Maybe it is impossible to create a perfectly uniform magnetic field that increases with $t^4$? But if so, why would this be impossible?
  2. Although the electric field is constant (at least per my reasoning) at any instantaneous moment, that constant value will change over time since the $E$ field is proportional to $t^3$. Maybe the time dependence of the $E$-field means something?
Elio Fabri
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heroes charge
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    How you know that there is a solution of Maxwell's equations describing this type of magnetic field you are talking about? Actually time dependent magnetic field means that there is electromagnetic field in the form of electromagnetic waves. – Alex Trounev Nov 24 '21 at 03:38
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    The divergence and curl only specify a vector field uniquely (using the Helmholtz decomposition) when the fields decay fast enough at infinity. I think what is happening is that because your field does not die off at infinity, your problem doesn't have a unique solution. I suspect you will actually find an infinite family of possible solutions to Maxwell's equations with a cylindrical electric field as in @mmesser314's answer, and with the center of the induced $E$ field a free parameter. – Andrew Nov 24 '21 at 05:02
  • Related question: https://physics.stackexchange.com/questions/194136/gausss-law-in-a-uniform-charge-distribution-extending-infinitely-in-all-directi/ – knzhou Nov 24 '21 at 18:55

3 Answers3

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The left side of the equation will not be $0$. You are being misled by how symmetric the B field is.

One place you might expect to find a uniform time-dependent B field is inside a solenoid. This is a cylindrical region of space, with a current flowing around the cylinder. If you just think about how symmetric the B field is, you might expect no current could generate such a B field.

Likewise, the induced E field is cylindrical.

enter image description here

Picture from https://faculty.uml.edu//Andriy_Danylov/Teaching/documents/L18Ch33InducedEcovered.pdf

mmesser314
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  • I see, I am still not understanding why my reasoning that every point in the space is indistinguishable, hence the electric field should be the same everywhere. In the case of the solenoid, it seems like the solenoids provides a cylindrical symmetry, hence the circular electric field lines, however in the case where there is empty space, there is no solenoid that provides a reference point, and there is symmetry at every single point? – heroes charge Nov 24 '21 at 04:35
  • @heroescharge and how would you be creating a time varying uniform field in empty space? – anna v Nov 24 '21 at 05:25
  • @annav I am not sure how one would produce such a field, which is why I was thinking that maybe the answer is that such a field cannot exist without other things such as a solenoid coil. – heroes charge Nov 24 '21 at 05:50
  • @heroescharge if the boundary condition for the problem to solve the equations is "emtpy space" the solution must be zero from the mathematics, imo. i,e the field you propose is not a solution of the equations. – anna v Nov 24 '21 at 06:19
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    The solution of Maxwell's equations is only unique if the fields fall off sufficiently fast at infinity. Since your field does not, the solution is not unique, and I suspect there is actually an infinite number of solutions parameterized by the center of the "solenoid" that @mmesser314 is describing in their answer (eg, the circulating electric field can circulate around any point). – Andrew Nov 24 '21 at 12:28
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However, since the magnetic field is uniform, any two points in the space are indistinguishable, so I believe that the electric field must also be uniform everywhere in that region of space.

This is not so. Any two points are not indistinguishable just because they have the same magnetic field. They are still distinguishable in that they have different position in space.

In a realistic scenario, magnetic field can be made only approximately uniform in a finite region of space. Then the points of space differ in how far they are from the source of the magnetic field or from boundary of that region.

In purely hypothetical case of magnetic field uniform in the whole infinite space, we can only conclude that curl of electric field is uniform. But this does not in any way imply electric field is uniform; on the contrary, it can't be, because then curl would vanish, as you have realized.

Ján Lalinský
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A uniform magnetic field means that $\vec{B}(r,t)=\vec{cst}$ ,i.e $\frac{d\vec{B}(x,t)}{dt}=\vec{0}$

The Tiler
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  • This is not the standard usage that I'm familiar with. "Uniform" means not varying in space; "static" means not varying in time. – Michael Seifert Nov 24 '21 at 19:15