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Given an infinitely long wire, carrying a linearly increasing current, what is the shape and direction of the electric field induced by the magnetic field?

Given increasing magnetic flux in a cylindrically symmetric region, the shape of the induced electric field is circular. In this case, using the Faraday-Maxwell equation, and rectangular loops in 3 mutually perpendicular planes, we can prove that the resultant electric field is composed of straight lines, and is directed opposite the current.

However, induced electric field lines always form closed loops. I would greatly appreciate it if someone could diagrammatically show me the individual closed loops (with direction), and explain how they vectorially add to form the resultant field..

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The problem that you have is that your wire is infinite. If you apply the reasoning that you used to arrive at the result that the electric field lines always form closed loops to the currents, you would arrive at the requirement that currents should always form closed loops. Your current does not. Mathematically you are requiring the divergence of the electric field (or the current density) to be zero in a finite region.

To see why the electric field lines don't form loops for the infinite case, you can imagine the wire either 1) long and finite, or 2) circular but with a enormous radius like the radius of the earth. If it is long and finite, it takes charge from one end and dumps it at other. The electric field lines will begin and terminate on those charges. If it is long and circular, your current now is a closed loop. Your electric field lines also form closed loops. Your straight line field lines now form large circles equidistant from the wire at all points.

For any finite wire, for which your infinite wire is an approximation that makes the calculation easier, the electric field lines will close if there are no charges and at least some of them will terminate on charges if there are charges.

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  • $\begingroup$ Very well stated $\endgroup$
    – Andrew
    Apr 26, 2021 at 15:45
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. . . . . the shape of the induced electric field is circular . . . . .

As you can choose any loop when apply Faraday this is incorrect as your example with a rectangular loop shows.

Consider a rectangular loop whose plane also contains the infinite wire carrying an increasing electric current, $I$.

enter image description here

To the right of the wire the magnetic field $B$ is increasing into the screen.

If the loop rectangular loop $abcd$ was made of a conductor an induced current, $i$, would be produced - Lenz.
That current direction gives you an indication of the direction of the electric field which is induced in the loop which does not necessarily need to be a conductor or even that shape.

. . . . . we can prove that the resultant electric field is composed of straight lines , and is directed opposite the current.

I am not sure what you mean by this.
Faraday tells you whether an emf (related to the induced electric field) is induced in a loop and about its magnitude but not how that emf is distributed within the loop.

So you could imagine the rectangular loop getting bigger an bigger so you only "see" part of the $ab$ section of the loop but the rest of the loop is still there.

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