We have checked out magnetism in a big way the past couple weeks. We know that magnetic fields are strange and circulate around moving charges and currents. We know that magnetic forces are created by magnetic fields acting on moving charges and currents: F = qv x B and F = Il x B.
Because these forces are cross products, moving charged particles will be put into circular motion by the magnetic force, and we have used the flat-hand right-hand/left-hand rules to figure out the direction of the push. We have also seen a major application of these forces in velocity selectors and mass spectrometers.
We then worked on the fields and forces created between multiple parallel wires with currents. This is a nice application of combining Ampere's law with the RHR's and the magnetic force equation, F1 = (I1)l x B2 and F2 = (I2)l x B1.
We then got into the production of magnetic fields, using Ampere's law for three special, symmetric cases: long straight wires, long solenoids, and toroids. This is built upon the notion of a path integral, since magnetic fields follow either circular paths (straight wire and toroid) or a linear path (inside solenoid). With the 'long' approximations, we do not need to use calculus, and it is B*(length of path) = mu*I_inside.
Then we hit the most difficult portion of all this, with the Biot-Savart law. This is used to find the magnetic fields for every other case, and we focus on three: single moving charged particles, a loop of current (like in our lab) and a straight wire with ends. We treat these cases as they really are - a bunch of moving point charges, where we add up all the little B-fields to get the total B-field, using integration. We also can use this to find the effects of multiple wires.
We even got into the worst-case example for Ampere's law of a NON-uniform current density in a wire, where we need to integrate to find the current inside a certain portion of the wire.
One last piece of the puzzle is the Hall effect. This is a phenomenon that happens when a material carrying a current is placed in a B-field (directed at an angle to the current flow), and the material is polarized due to the magnetic force on the current. That polarization of the material can be measured as a voltage difference, and if the material is known, the magnetic field strength can actually be measured (a so-called Hall probe).
There's quite a bit here for plain magnetism, so hopefully these videos are useful!