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

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

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

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.”