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I am trying to understand how to determine the shape and direction of the electric fields induced by a changing magnetic field. But all of the resources I have read make use of the following symmetry argument which I don't understand:

Consider a magnetic field increasing in magnitude at a constant rate. The magnetic field can be generated by an ideal solenoid. Then, by way of the symmetry argument, the electric field induced by this magnetic field will be circular in nature.

I'm not sure what is meant by symmetry here. Does it mean:

The electric field set up will have to be circular because the magnetic field within it uniform in magnitude and distribution (at a differential time dt), thus to oppose this flux, there has to be a symmetrical magnetic field created by the induced electric field.

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The magnetic field of an infinite solenoid has cylindrical symmetry. That is, it literally looks like a cylinder

For any given coordinate $\theta$ the distribution "looks" the same.

This means that the field value for all $\theta$ will look the same (provided the homogenous solution is zero).

From the definition of the curl it is clear to see the direction.

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  • $\begingroup$ I have read about the definition of the curl, but I'm not sure of whether it describes the origin of the shape of the electric fields or if it was developed after the shapes have been determined in the first place. Would appreciate if you could clarify this. $\endgroup$
    – Piksiki
    Commented Jul 31, 2022 at 16:33
  • $\begingroup$ The curl of the electric field is dependant on a changing magnetic field. We know the curl as we know the magnetic field. so this determines the shape of the field. Although "what comes first" doesn't really matter $\endgroup$
    – jensen paull
    Commented Jul 31, 2022 at 18:20

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