Ampere's Law was discovered in 1819 by Ampere. Ampere's Law states that currents generate magnetic fields or in other words, whenever you have a current, there is a magnetic field circulate around it. Ampere's Law is expressed in the equation magnetic field x 2(pi) x radius = constant x current(passing through that path).
So let's talk about ampere's law. Ampere's law was discovered well the effect behind ampere's law was discovered around 1819 actually by a Physics lecturer who was in the middle of a lecture and he noticed that a magnetic field was generated that moved around a compass needle when he ran a big current through a wire, so the law basically says that currents, if I run a current it will generate a magnetic field. There is multiple ways of stating this law; there's the [IB] law, there's oersted's law. The standard way to present it is in terms of ampere's law which basically says that whenever you've got a current, you'll have a magnetic field that circulates around that current. Now in order to determine the sense of that circulation what you do is you use your right hand and you grab the wire with your thumb pointing in the direction of the current. When you do that, your fingers will show you the sense in which the magnetic field circulates around so if you've got the current going that way the magnetic field is going to make little circles around the current in this sense so in the front it will be going down in the back it will be going up beneath the wire it goes into the board and above the wire it goes out of the board so that's the idea.
Now what ampere's law also says is that the magnetic field circulation and by that we basically mean the magnetic field times the length of that circulation is proportional to how much current we've got so we double the amount of current then we're going to double the size of the magnetic field alright this also gives us a very very very easy way to determine a closed form expression for the magnetic field due to a long wire so here I got a wire where the current coming out of the board so that's why you don't see the wire because the current's coming out of the board. Now the magnetic field is the blue line notice that again I grab the wire with my thumb pointing in the direction of the current and my fingers indicate how the magnetic field circulates.
Now ampere's law says that the magnetic field circulation which is the magnetic field times the length of this curve which is 2 pi r because it's our circle is proportional to that means that it equals a constant times the current that's coming out of the board. Now what is this constant? Well we could actually measure it but in reality what people do is they use the freedom that we have to define the ampere as a unit of measurement in order to make this constant exactly 4 pi times 10 to the minus 7 in S.I units so those units are Tesla, meters per amp alright so 4 pi times 10 to the minus 7 that mue zero it's called the permeability of free space. Alright so when I've got this expression I'm going solve for B and it gives me B equals mue not I over 2 pi r so that means if you double the distance that you are from the wire, you're going to have the magnetic field. If you triple the distance from the wire, you cut it in thirds.
Alright, let's do an example with this result, so I want to know what the magnetic field is 2 centimeters away from a wire that carries 5 amps of current alright why don't I just do this directly? B equals mue not I over 2 pi r so I'll just plug in 4 pi times 10 to the minus 7 times 5 amps divided by 2 pi times alright what we know we need to do we need to change it to S.I units also our magnetic field will not come out in Tesla so 2 times 10 to the minus 2 alright this point, wonderful things happen, look at this 4 pi, 4 pi, pi pi alright so now we end up with 5 times 10 to that negative seven plus 2 is negative 5 Tesla alright or 50 microtesla alright so that's the magnetic field it's not a really big magnetic field and 5 amps is a fairly large current so when we do this just with the long straight wire generally we don't end up with a huge magnetic field well I have other ways to do that later.
Alright suppose that I want to determine the force between two long wires, so here's the idea got two long wires they're both carrying current. This long wire oop I use the right hand rule and that means that there is a magnetic field into the board here up the second wire alright but now a wire that carries current in a magnetic field it feels a force what is that force? Well il cross b so that force is actually attractive in this case it's going to pull the wire up towards it well a big is that force, well geez, the magnetic field a distance d away from wire with current II 1 is mue not I 1 over 2 pi d okay.
The force is the second current times the length of wire times the magnetic field which I can rearrange like that. Now usually what people do is they'll divide by the length and they'll get a force per unit length that's given by this formula. Notice that we got mue not we've got the product of the two products and then we divide by 2 pi d. Now this is actually how the ampere is defined. The amp is the current that gives exactly a force per unit length of 2 times 10 to the minus 7 Newtons per meter when you have two wires carrying the same current and they're 1 meter apart so that means that this is 1 this is 1 this 1 and I need the whole thing to be 2 times 10 to the minus 7 so that means is 4 pi times 10 to the minus 7 and that's where the definition of the ampere comes from and that's ampere's law.