Magnetic field on a loop due to another loop

[aldo sebastian]

In the attached picture, the outer wire is carrying a current I(t), and it's asked to find the induced EMF in the inner loop. Now I have indeed calculated the B-field along the z-axis due to the outer loop. My lecturer then puts z=0 into that expression and then multiplied it with the area of the inside loop to get the flux and then thus EMF. My question is, the magnetic field component into the inside loop is obviously not only the one on the z-axis; the off-axis B-field that is inside the loop should also be taken account right? Or is there some symmetry that I am missing? Thank you

 

[BvU]

$B_r$ is indeed zero at z=0, folllows from Biot-Savart: the current is in the xy plane so the field is perpendicular, i.e. in the $z$ direction.
(see e.g. here for a formula). So in that respect teacher is correct.
Using $B_z(r=0)$ for $r > 0$ is much more questionable -- see the formulas -- but rather hard to quantify. There's a picture here (28 nov 2015)

 

[rude man]

In the attached picture, the outer wire is carrying a current I(t), and it's asked to find the induced EMF in the inner loop. Now I have indeed calculated the B-field along the z-axis due to the outer loop. My lecturer then puts z=0 into that expression and then multiplied it with the area of the inside loop to get the flux and then thus EMF. My question is, the magnetic field component into the inside loop is obviously not only the one on the z-axis; the off-axis B-field that is inside the loop should also be taken account right? Or is there some symmetry that I am missing? Thank you

The B field can only be accurately determined on the z axis including at the center (and you have to assume an infinitely long wire). The B field in the area of the inside loop other than on the axis is extremely difficult to determine. It involves elliptic integrals, is always an approximation, and even involves the diameter of the wire. So saying the flux is the axial B field multiplied by the area of the inner loop is a rough approximation, yes.