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Say we have this situation: enter image description here where there's a magnetic field inside an imaginary circle with radius r (which means there is no wire!). There's a change in magnetic field over time which means that the flux is also changing. Say we want to calculate the electric field in any point in space, with radius bigger or smaller than r.

My first guess would be to use faraday's law:

enter image description here

But isn't that law just for a circuit/wire? I know that the left side of the equation is just $$ \begin{align} 2\pi ER &&\text{(not r)}\ \end{align} $$ and from there we would be able to calculate $E$. But is this valid?

Aditional question: say that circle is now an actual copper wire for example, and we use that law to calculate the EMF. Will there still be an electric field outside that wire too? I guess so but I want to be sure.

Thank you in advance.

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I know that the left side of the equation is just 2πER(not r) and from there we would be able to calculate E. But is this valid?

It would be valid, only if E was the same everywhere around the circle, which would be the case if the magnetic field was symmetric relative to the circle, for instance, if the circle was concentric with a current loop producing a changing magnetic field.

It would not be valid, if E was not the same everywhere around the circle, for instance, if the circle was not concentric with a current loop producing a changing magnetic field.

Additional question: say that circle is now an actual copper wire for example, and we use that law to calculate the EMF. Will there still be an electric field outside that wire too? I guess so but I want to be sure.

In the absence of a perfect shielding, an electric field will be everywhere.

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Faraday's law is valid for any surface.

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