EPT
When electricity flows in a wire, it produces a magnetic field. You can easily look up pictures online of the direction of a magnetic field around a straight wire. Calculating the magnitude is a bit more tricky.
The most basic way to calculate he magnetic field is to use the Bio-Savart law, which is basically just the integral you have to solve to get the magnetic field for a 'continuous current distribution.' It's the analogue to using E = integral[k*dq/r^2] to find the electric field near a continuous charge distribution. The problem is that this integral (bio-savart) is ugly and/or impossible to solve for most problems aside from the easy ones.
During a chapter on static electricity you learn about how you can use Gauss' Law instead of the big integral associated with Coulomb's Law for solving some problems that have a high degree of symmetry. This is often a great shortcut. The analogue to that for magnetic fields is that you can use Ampere's Law to find the magnetic field near currents (provided the appropriate symmetry and whatnot). For Gauss' law you imagine a closed surface wrapped around a charge distribution and relate the electric flux passing through the surface to the amount of charge inside the surface. For Ampere's law instead of creating a closed surface you create a closed curve.
Way to think about a closed surface: an empty plastic bag that is stitched closed to have no opening with a charge distribution placed inside it.
Way to think about a closed curve/loop: a string with both ends tied together to make a circle or similar shape. This actually creates an 'open' surface. Think of a normal plastic bag with a draw-string at the top/opening. In order for current to pass through the bag it needs to pass through the loop of the string AND it needs to bust through a hole in the bag somehow (similar to how electric field lines pass through the closed surface) otherwise the current would get trapped by the bag, which would be pretty depressing for the current since it just got trapped by a bag which doesn't actually exist.
Of course for Gauss' and Ampere's Law there is no actual surface or loop near the charge/current distribution... it is just an imaginary construct placed to allow us to use the simple formulas. To use Ampere's law you place the closed curve around the current so that the string lines up parallel to the magnetic field. The current passes 'through' the loop of the string, like a beam of water being shot through a ring of fire... or something like that. Now you can mathematically relate the amount of current passing through the loop to the strength of the magnetic field where the 'string' of the loop is. If you want to know the magnitude of the magnetic field near the current you need to make a loop with a small radius; if you want to know the magnetic field further from the wire you need to make a loop with a bigger radius.
Ampere's Law, u0*I = closed_int[B*dl] works great for simple examples such as a long, straight wire. There is a flaw, however, that Maxwell discovered (and integrated into his more complete 'Maxwell's Equations')
Imagine a capacitor. Current flows through a wire into one plate of the capacitor, and then an equal current flows away from the other plate through another wire. Since there is current flow there is a magnetic field, and while the capacitor is still mostly uncharged it behaves mostly like a closed switch. Let's attempt to use Ampere's Law to find the magnetic field produced by the wire leading into the first plate. We take our closed curve (string) and wrap it around the wire a fixed distance (radius) away. You basically just have to do [u_0 * I / circumference_of_loop] in order to find the strength of the magnetic field. But Maxwell discovered a way to make this 'break.' He placed his closed loop in the same place but imagined the open surface it created (open plastic bag) to be positioned so that the first plate of the capacitor was inside the bag. The bottom of the bag was positioned between the two plates so that no hole needed to be punched in the plastic bag. Looks sorta like this:
![[image loading]](http://img97.imageshack.us/img97/6068/driftcurrentcircuit.jpg)
Image Courtesy of Glider*
You can see that the closed loop (orange) wraps around the current without any problem, but the open surface (green) it creates is stretched so that the wire doesn't cut through it (this is a 2d depiction but is really a 3d situation so beware). If the current doesn't cut through the open surface then according to Ampere's Law there should be no magnetic field... but if you choose to use a smaller 'plastic bag' then the wire will cut through the bottom of the bag. How could the size of the imaginary bag affect how much current passes through the closed loop? Maxwell realized that Ampere's Law was wrong. His solution? Add a new term to the equation until it works for all situations regardless of how the open surface is chosen.
The Displacement Current: I_d = e_0 * d E_flux / dt
The fact that charge was building up on the capacitor plates means that an electric field is being produced steadily... and you have to take that into account when trying to use Ampere's Law to calculate the induced magnetic field. In Ampere's Law, where 'I' appears, Maxwell said substitute in I+I_d where 'I' is the current being carried by a wire/etc, called the conduction current, and I_d is the displacement current which only comes into play when there is a changing electric field penetrating the open surface of the closed loop.
This has been just a brief attempt to make clear some aspects of how to calculate magnetic fields being induced by currents before and after Maxwell added the Displacement Current to Ampere's Law. There was no major attempt to help the reader understand the mathematics behind using Ampere's Law to begin with (it's simple to solve algebraically, but a bit difficult to set up perhaps). Since each reader has a different background with this material it will probably come across as confusing (vocab, assumptions, etc) for some people and elementary and obvious to others... but oh well. If any notation, names, etc are confusing please ask!
*Just kidding. Glider couldn't make an image look this bad if he tried :p
