# Flux through circular loop due to long wire at center

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?

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.

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