As we discussed earlier, our canonical example with be the long straight wire. In this case, the wire is a bit thick, so that we can investigate what happens inside the wire as well. That's a tough job with the Biot-Savart law. We will use Ampere's law to find the magnetic field both inside and outside the thick wire. On this page, we will focus on the right-hand side of Ampere's law, namely, the current through our Amperian loop.

Ampere's Law relates how curly the magnetic field is to how much current produces it.

$$\oint \vec{B}\cdot d \vec{l} = \mu_0 I_{enc}$$

The right hand side describes the amount of current enclosed by the Amperian loop. That is, how much current runs through the area of the loop face. The figure below describes relationship between the loop and the enclosed current.

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For the purposes of these notes, let's assume with have a thick wire with total uniform current, $I_{tot}$.

In many cases the radius of the loop will be larger than the radius outside of a wire, that is, it will enclose all the current. This is the simplest of cases where $I_{enc} = I_{tot}$. This will be the case for finding the magnetic field outside the wire.

In some cases, the radius of the loop will be smaller than the radius of the wire. In that case, you will enclose some but not all of the current $I_{tot}$. To find $I_{encl}$, you will need to know the current density $J=I/A_{wire}$. For our purposes, we will assume a constant current density because we will deal with uniform currents. Hence the enclosed current is a fraction of the total,

$$I_{enc} = J A_{enc} = I_{tot} \dfrac{A_{enc}}{A_{tot}}$$

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where both of these areas are cross-sectional areas. $A_{tot}$ is the total cross-sectional area of the wire and $A_{enc}$ is the cross-sectional area enclosed by the loop. This is very similar to how you found $Q_{encl}$ with Gauss's Law for Electric Fields.