1. Cross-section of the wire:
Example: a cylindrical wire of radius R carries a current I that is uniformly distributed over the wire’s cross section. Calculate the magnetic field inside and outside the wire. I R
Cross-section of the wire: 
direction of I B
Choose a path that “matches” the symmetry of the magnetic field (so that the dot product and integral are easy to evaluate); in this case, the field is tangent to the path. R r

2. Over the closed circular path r: 
direction of I B R r
Combine results and solve for B:
B is linear in r.

3. Outside the wire:  R r (as expected). B Plot: B r R direction of I
A lot easier than using the Biot-Savart Law! r
(as expected). B
Plot: B r R

4. Calculating Electric and Magnetic Fields
Electric Field
in general: Coulomb’s Law
for high symmetry configurations: Gauss’ Law
Magnetic Field
in general: Biot-Savart Law
for high symmetry configurations: Ampere’s Law
This analogy is rather flawed because Ampere’s Law is not really the “Gauss’ Law of magnetism.”