The interactions between electric currents and magnetic fields create forces. The magnetic force on a current in a magnetic field = current x displacement across the magnetic field. This force can be predicted using the right hand rule. When two currents are directed through magnetic fields in the same direction, they attract. When they are directed in the opposite directions, they will repel.
Alright let's talk about the interaction between currents and magnetic fields. Now there's two major types of interaction. We have just a current that's in a magnetic field it will feel a force on it and that's the first one we're going to talk about and then there is another interaction which is associated with the magnetic field that the current itself will generate and that's separate we'll talk about that in a minute.
Alright, so first of if I've got a current in a magnetic field, well current consists of a bunch of moving charges and any time I've got moving charges in a magnetic field, I've got a force so let's see if we can relate this force using the Lorentz force law to the current. Well we'll have charge velocity cross magnetic field now what we're going to do is we're going to think of the velocity as displacement over change in time and then we're going to slide this change in time underneath the charge. Now what's nice about that is that charge over time is current so that tells us that the magnetic force on a current in a magnetic field is just the current times the displacement cross the magnetic field. Alright, so let's see this in action. Alright, magnetic field directed into the board we have a current going up and I want to know what direction is the force in alright well it's the delta r the displacement which is a vector that just goes up like that from where it enters to where it exits the field there like that the force is to the left.
Alright, now suppose I've got a situation like this one where the wires are curved? Well delta r as long as the magnetic field is constant, delta r is a vector from where the current enters the field to where it exits so this will be delta r so now all I need to do is put my thumb in the direction of delta r, my fingers in the direction of the magnetic field and my palm points in the direction of the force. Notice that it's kind of a combination between what it would be if I had a just straight up and what it would be if I just had it horizontal alright it's a combination it's a vector sum of those two things.
Alright, let's do an example problem so I want to know the force on 50 centimeters of wire carrying a current of 5 amps upward in a 3 Tesla magnetic field out of the board. Alright well first let's do directions, 3 Tesla out of the board there it is right? I've got the current going up and so will do thumb, fingers the force is directed to the right alright. What is that force? Well f equal ILB, I is the current, L is the length of the displacement vector, now we can only use this if the displacement, with a current, and the magnetic field are perpendicular otherwise we'd need the part of that that was perpendicular but most of these problems it's already perpendicular, so we'll do current. What was that? 5 amps the length is 50 centimeters but of course we've got to work in SI units so we'll write point 5 alright? And then we've got the magnetic field 3 Tesla 5 times point 5 is two and a half, two and a half times 3 oh jeez, three quarters is 75 cents so this is going to be 7 and a half Newtons and there you go it's just that easy only thing that sometimes you might have to worry about is that maybe the current's not perpendicular to the magnetic field and then you just got to take the component that is perpendicular shouldn't be easy to shouldn't be difficult to find.
Alright let's look at the second case. Now this is somewhat different so force between wires. Alright if I've got two wires here that are both carrying current in the same direction, they're going to put a force on each other why? Well because this top wire generates a magnetic field just because it's carrying a current so that magnetic field I get using the right hand rule by grabbing that top wire with my thumb in the direction of the current then look down below the wire the magnetic field is directed into the board so that means that this current, because it's in the proximity the vicinity of this other wire, is sitting in the magnetic field that's directed into the board alright boom, boom look at that! There's a force that acts upward on that bottom wire so that means that if two currents are in the same direction, the magnetic force between them will be attractive. Now we can actually understand this directly from a magnetic field diagram, so here's two currents they're both coming out of the board that means they're in the same direction right? So I want to know what does the magnetic field looks like for both of these guys. Well here we go I'm going to grab the wire with my thumb pointing in the direction of the current and my fingers are going to be the magnetic field so the magnetic field this is going to circle around like that.
What about here? Same thing alright there's the next line, the next line and now as we get bigger notice that in the middle this magnetic field is going down and that one is going up so that's going to allow them to cancel so we get these weird goggles looking figure alright and then we keep on going and look at that you can see attraction in this diagram because the magnetic field can cancel in the middle and that means that it's not as strong in the middle alright? And that's going to attract the two wires together alright. Let's see what happens when we change the direction of this bottom current so now it's going the other way. The magnetic field is still the same because I didn't change the top current but now look what happens, when I do the right hand rule I get a repulsive force so when the two currents are in the opposite direction, I get repulsive so that's kind of weird because with charges opposites attracts but with currents, opposites repel so let's see how this goes we've got the repulsion down here and let's see what the magnetic field line's going to show us so we've got into out of, alright let's draw these. Now here I got to put my thumb in so that means that I'll get circles but they'll be going the other way that does, well in the middle notice that they can't cancel anymore so that means that as I keep drawing them they're going to spread out over on the other side but I can't make that goggle figure anymore because now all these field lines are pointed the same way so this is repulsion alright so when currents are directed in opposite directions, they repel same direction they attract this actually was what led to the definition of the ampere.
An ampere is the amount of current that flowing through two identical very long wires that are parallel held one meter apart will feel a force of 2 times 10 to the minus 7 Newtons and that's actually the definition of an ampere, kind of strange but that's the interaction of currents with magnetic fields.