1. Ampere’s Law
Just for kicks, let’s evaluate the line integral along the direction of B over a closed circular path around a current-carrying wire. I B ds r
The above calculation is only for the special case of a long straight wire, but you can show that the result is valid in general.

2. Ampere’s Law
I is the total current that passes through a surface bounded by the closed (and not necessarily circular) path of integration.
Ampere’s Law is useful for calculating the magnetic field due to current configurations that have high symmetry.
The current I passing through a loop is positive if the direction of integration is the same as the direction of B from the right hand rule. I I
positive I
negative I B ds B ds r r

3. Your starting equation sheet has
Your text writes
because the current that you use is the current “enclosed” by the closed path over which you integrate.
Your starting equation sheet has
The reason for the 2nd term on the right will become apparent later. Set it equal to zero for now.
If your path includes more than one source of current, add all the currents (with correct sign). I1 ds I2