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The question asked was:

A long straight current carrying wire passes normally through the center of a circular loop. If the current through the wire increases, will there be an emf induced in the loop?

The answer I wrote was:

Since the loop is assumed to be an ideal thin wire, the cross sectional area tends to zero. Also, since there is negligible amount of length for emf to be induced, there is no flux linkage with the loop and therefore any change in current through the wire produces mo effect on the wire or emf.

I think that my answer is incomplete and assumes too much. The actual reason should be far more general than what I wrote, but I don't understand the entire concept of flux in this case since the magnetic fields forming closed loops is different from the intuitive version.

Any clarification will be appreciated.

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The magnetic field is perpendicular to the current's direction, and the current is going normal to the loop, meaning that $B$ is parallel to the loop's plane. Because flux is the dot product of the magnetic field and the loop's normal "surface vector", and the angle is $90^\circ$, it is equal to $0$, thus not changing in time and not inducing an emf.

$$ \varepsilon = -\frac{d \Phi}{dt} = -\frac{d}{dt} (0) = 0 $$

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According to me This concept is idealistic. Induced emf may be induced due to relative motion between the electrons. Force experienced by electrons in a loop is F=Bqv. Since the current is changing in the wire and hence the magnetic field at the electron in the loop will induce the emf induce in the coil. Here field is time dependent and is responsible for inducing the emf in the loop.

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