# Does magnetic intensity ($H$-field) 'conduct' through a material

We've been taught by our lecturer that magnetic intensity propagates through a material. But I have a doubt because I feel that it is not consistent with Ampere's law. My reasoning is explained below. The picture above illustrates how we were taught to apply Ampere's law, essentially leading to $$H=\frac{Ni}{L}$$ where N, is the number of turns, i is the current and L is the length of the nickel frame.

However, my problem with that is I thought using Ampere's law you could take any loop, so the picture shown below would also be valid: However, this method seems to suggest that the H-field does not propagate through the material, and is in fact zero outside the coil, and $$H=\frac{Ni}{L}$$ inside the coil. But this method does also not seem to be consistent because we are also taught that $$B = \mu H$$ which would suggest that the magnetic field (B) is zero outside the coil also.

Either way there seems to be a contradiction somewhere in what I've been taught and I was hoping someone could clarify all this for me.

• Outside the coil $\mu = \mu_0$ do how do you conclude that $B=0$? – my2cts Aug 21 '19 at 6:42

$$H$$ field is not exactly zero outside the core, if it was, the loop integral over the second loop would have non-zero value, despite the fact there is no conduction current there.
The integral over the second loop is zero, but this does not mean $$H$$ itself is zero on that loop. Contributions to the integral from the left side and right side are roughly equal magnitude but opposite sign so together with the contribution from top and bottom they cancel each other.