 So now let's take a look at the equation for the magnetic force on a current carrying wire. I'm going to start off by looking at the equation we had for the force on a charged particle moving through a magnetic field. And if we wanted to find just the magnitude of the force, it was equal to the magnitude of the charge, the magnitude of the velocity, the magnitude of the magnetic field, and the sign of the angle in between the velocity and the magnetic field. In order to start working this equation, we're going to start by taking a recognition here of this speed. And this speed is really the distance traveled over a certain amount of time. So for a single charge particle that's moving along a straight section of wire, we can replace that velocity by the distance traveled and the time. Now, for right now, I'm going to go ahead and drop all of my magnitude signs. And the reason I'm dropping that is just to try and keep everything a little bit neater as I'm doing the algebra. So I'm going to take this equation here where I've substituted in for the velocity for the distance over the time and start doing some algebra on that particular equation. So here's my equation again. And I'm going to take a look at this quantity here of the q times d over t. Algebraically, that's exactly equal to pulling the q inside those parentheses, the q and d and the numerator and the t and the denominator. Now I'm going to take the fraction here and I'm just going to rearrange it just a little bit. So I recognize that that's the charge divided by the time multiplied by the distance. So q over t times d is exactly the same as qd over t. So I'm going to take this equation which is just rearranging my original equation and think about it a little bit differently. So I'm going to start by looking at this quantity q over t. And a recognition that q over t physically is the current that's flowing. And if I've got a wire, the distance is really just the length of the wire. So I don't have a single charge moving over a distance in a particular amount of time. I've got a set of charges moving at a particular rate through a length of wire. By making these substitutions, my new equation tells me that the force is equal to the current times the length times the magnetic field strength times the angle in between the wire and the magnetic field. It's important to remember though, originally I had dropped all of those magnitude signs just to make the algebra easier. So far all I have found is the magnitude of the force. I can think of current as being how much current I've got, the length of the wire as being just a scalar length, the magnetic field, and my sign of my angle between it. So I expect all of these to be positive quantities and that's going to give me my magnitude of the magnetic force. It's really easy to start getting things mixed up when you're talking about the magnitude of the magnetic force. But just try and keep your word straight. It's easy to get tongue-tied over that. Now because that's just the magnitude of the magnetic force, if I want to get the direction, there comes my right-hand rule again. We're going to do a separate little video clip to explain that. But what if I did want to get my direction and I didn't want to have to use the right-hand rule? Or maybe I'm in a situation where they don't tell me the angle directly. Well just like we had our force on just a charged particle, this is where we end up getting our cross-product form. So rather than using the magnitude form, I can start to think about the sine theta as the angle in between my two vectors. Well obviously the magnetic field is a vector and so that's going to be one side of our cross-product. But I've got this i and this l and I just said that i is just how much charge and l is just the length of the wire. So how am I going to represent one of those as a vector? There's two different ways to think about it. One is to think about the current as having a flow direction. So it's not just how much current is flowing but also which direction the current is flowing in. And so you could represent the current as a vector quantity. More commonly though, we think about the current as being how much current is flowing and recognize that in a wire it is forced to flow along the length of the wire. So I can represent the geometry of the wire as a vector quantity, which direction is the wire pointing and that gives me a way to get my vector for my length. Regardless of which one of those you choose, we then have the vector force. It's not just the magnitude but the i, j, k vector components for which direction is the force in. So in all, we've got two different equations we can use. One if we only care about the magnitude of the force and particularly if we're given the direction between the wire and the magnetic field as the angle. And the other one if we're trying to find the full-blown vector force especially if we're given the length and the magnetic field as vectors.