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Example when an current carrying loop is placed in external magnetic field varying with time some emf is produced in the coil now we know that emf is due to electric field and the loop is experiencing some current so there must me some electric field responsible for it .Is the electric field present here cricular non-conservative?

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  • $\begingroup$ en.wikipedia.org/wiki/Betatron $\endgroup$
    – John Doty
    Sep 17, 2022 at 14:29
  • $\begingroup$ Can you please elaborate this a bit about non conservative fields essentially circular ones $\endgroup$ Sep 17, 2022 at 16:21
  • $\begingroup$ If an electron can go around in a circle, returning to its starting point with an increased energy, you have a non conservative field. $\endgroup$
    – John Doty
    Sep 17, 2022 at 16:32

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Yes, the electric field $\mathbf{E}$ induced by a time-varying magnetic field $\mathbf{B}$ is according to Faraday's law of induction $$\mathbf{\nabla}\times\mathbf{E}=- \frac{\partial\mathbf{B}}{\partial t}$$ necessarily non-conservative. To see this remember the definition of a conservative field. An electric field $\mathbf{E}$ is called conservative if it can be derived from a potential $V$ $$\mathbf{E} = -\nabla V$$ or equivalently, if it is irrotational, i.e. $$\mathbf{\nabla}\times\mathbf{E} = \mathbf{0}$$

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The electric field in the potential formulation takes the form:

$$\vec{E} = -\nabla V - \frac{\partial\vec{A}}{\partial t}$$

In the coulomb gauge;

The term $-\nabla V$ is purely divergent. ie - is only determined by the divergence of the E field

The term $- \frac{\partial\vec{A}}{\partial t}$ is purely solenoidal. ie is only determined by the curl of the E field

Faraday-Maxwell equation:

$$\nabla × \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$

A changing magnetic field coexists with a curl of the electric field, ie the component of the E field that is purely solenoid.

From the previous analysis the component of the E field that is "created" by a changing magnetic field is:

$$\vec{E}_{induced} = - \frac{\partial\vec{A}}{\partial t}$$

Provided we are in the coulomb gauge.

This is the "type" of electric field that is due to the coexistence of a changing magnetic field.

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