 In this very short miniature lecture, I would like to discuss a particular kind of device called a solenoid. We've been looking at things like magnetic fields from wires straight or loops of wires, for instance, or a wire that's been bent into a circle. Without going into terribly much detail about exactly how you calculate the magnetic field from a solenoid, which I'll describe in a minute, I can motivate where the magnetic field for a solenoid comes from, why it looks the way that it does, and then how you can use it in other problems. A solenoid is nothing more than a single loop of wire placed next to other single loops of wire. So an ideal solenoid has a loop of wire carrying current. Next to it is another loop of wire carrying current with the planes of the loop oriented parallel to one another, and then another loop, and another loop, and another loop. So that's the ideal solenoid. How in practice do you make this? Well, to make a solenoid, all you have to do is take a long piece of wire and, for instance, start to curl it around a cylinder. So as you wind it around the cylinder, you make one turn of copper wire and then another turn of copper wire, and then a third turn of copper wire, and you keep going making as many turns of copper wire as you need in your solenoid, and you'll see in a minute why it is that you want to carefully choose the number of turns of copper wire in a solenoid. So a solenoid is nothing more than a coil of wire tightly wound. Now let's consider what happens when you run an electric current through a solenoid. For a moment, let's consider what happens when you have a current flowing through a single loop of conductor, a single loop of wire. As you can see here, the magnetic field looks just like the magnetic dipole field from a bar magnet. And in fact, as I mentioned earlier in the course, the simplest magnetic field configuration that we know about is the dipole configuration. And you can see why. If moving current is the primary source of magnetic field in the world, and the simplest form of moving current is simply current moving in a small loop or circle or something like that, the simplest magnetic field you can ever get is the one shown here. It's a dipole field, and this is why we don't see magnetic monopoles in nature, because at its heart, the origin of magnetic fields is the motion of electric current, and that motion requires the generation of something that looks like this, a complicated dipole field. Now imagine what would happen if we took a few of these loops and we placed them next to one another. So in this picture here, what we're looking at is we're looking at loops of wire. We're looking at it from the side now. We've sort of sliced through the loops of wire, and you can see the little iron filings in the picture illustrating where the magnetic field is pointing. The magnetic field is beginning to become very uniform in the space inside the loops, going from loop to loop to loop. Now outside, it still looks very complicated. It's looping and swirling, but inside of the area, inside the enclosed area of the loops, we see that the magnetic fields are adding up from one loop to the next to the next, and they're becoming very uniform. This is how you make a uniform magnetic field in practice. A solenoid is a great way to do that. You just take a wire. You make many windings of the wire around a form of some sort, a non-conducting form like plastic or glass or ceramic, and you wind up making many tightly spaced loops of wire. If you can run a current through the wire, you'll have current circulating like water through winding pipes going down the length of the solenoid. With every turn, the loops make magnetic fields, and in the center, they all add up. In an ideal solenoid where you ignore the edges of the solenoid, the first loop and the last loop, and you just concentrate on the magnetic field inside of the core of the length of the solenoidal cylinder, you get an extremely uniform magnetic field. This is the principle, for instance, of how one creates a very strong uniform magnetic field for use in a magnetic resonance imaging device. I showed you this already in class, but this is a computer map of the magnetic field inside of a MRI solenoid. Now, this is composed of many, many, many thousands of closely spaced windings of wire, and so you get a very precise and very uniform magnetic field of a very particular strength, especially at the central core, the central axis of the solenoid where the patient would be placed, and that's very important for the operation of the device. So what does the magnetic field then at the end look like inside of a solenoid? Well, it has a very simple form, and again, I'm not going to derive this for you. I'm just going to show it to you and explain each of the pieces, because we're going to need it for the next step, which is magnetic induction. The magnetic field inside of a solenoid, an ideal solenoid, simply is equal to mu naught, this constant that we already know about, the permeability of free space, times the number of turns of wire, that's the capital N, times the current that's being driven through the wire, so N times I gives you the total current flowing through every one of the loops sort of all added together, because those, each loop creates its own magnetic field, if you have N loops of wire each carrying current I, that adds up to the total magnetic field, and so this N times I essentially represents the active adding up all of the magnetic fields of the loops of wire, and then this whole thing is divided by L, the length of the solenoid. So big N is the number of turns of wire, I is the current, and L is the length of the solenoid. And you can see now how you can control the magnetic field by controlling the geometry of the solenoid. You could make a very tightly wound solenoid that's very short, and that would give you a very strong magnetic field inside of the solenoid. You could weaken the magnetic field by increasing its length, or you could reduce the number of turns of wire. Now when manufacturing a solenoid, it's very convenient to write this N over L as a new number, which we'll just write as a little N, and it's referred to as the number density, it's the number of turns per unit length. So for instance, if you buy a solenoid for some purpose, you don't buy it and ask for a solenoid that's exactly one meter long with so many turns. It's easier for the manufacturer to simply manufacture a certain density of turns per unit length, number of turns per unit length, and then sell you the solenoid, you can cut it down to whatever size you need in order to get the appropriate number of turns. So if you know the number density and you know the length that you want, you can simply cut it down to get the number of turns that you want and so forth. So this little N is a very convenient thing, it lets us then write the magnetic field as mu naught times little N times I. So this concludes the lecture on the solenoidal magnetic field. We'll now use the solenoid as an archetypal magnetic device in the conversation about magnetic induction.