Anderson Video - Energy in an Inductor

Professor Anderson
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Okay. Is there some energy that we can store in an inductor? Right, it's hard to push current through this inductor, that means that we've done some work to do it, there must be some way to store that energy in the inductor itself. Remember when we charged up a capacitor, we of course, had energy in that capacitor. How does this work with an inductor? Well, remember work W was equal to QV. Okay but V is really, in this case, like EMF. What is the electro-motive force that's going to drive these charges around? So if I think about delta W, how much work do I do by moving a delta Q through the system? It's that. There's a little bit of a technical point, you got to have a negative sign on there, okay, I don't worry too much about the negative sign. But we know what epsilon is for an inductor, right? It is self inductance L, which is a value times delta I over delta t. Okay, L is the value of the inductor and it's measured in henries, so you might have a millihenry inductor. Look what happens. The minus signs cancel out and we get L out in front, we get a delta Q over a delta t, and then we still have a delta I. But this is L times delta Q over delta T is I, and then we have a change in I. So when you do this properly with integration you can figure out what the energy in the inductor is, and it is simply this: one-half L I squared. This is similar to what we were talking about with capacitors, right? Remember the capacitor was one-half CV squared. Similar thing, but now we have one-half L I squared. But if you're generating energy in this inductor, there really must be energy in those B fields themselves, because the B field in the center of the solenoid which is really important for the inductor, and so the energy in the B field itself it is the following. And we're going to write down the energy density which is energy per volume, and it is one over two mu naught B squared. Remember, for the electric field we had one-half with an epsilon naught times e squared, now we've got one over two mu naught times B square.
Okay. Is there some energy that we can store in an inductor? Right, it's hard to push current through this inductor, that means that we've done some work to do it, there must be some way to store that energy in the inductor itself. Remember when we charged up a capacitor, we of course, had energy in that capacitor. How does this work with an inductor? Well, remember work W was equal to QV. Okay but V is really, in this case, like EMF. What is the electro-motive force that's going to drive these charges around? So if I think about delta W, how much work do I do by moving a delta Q through the system? It's that. There's a little bit of a technical point, you got to have a negative sign on there, okay, I don't worry too much about the negative sign. But we know what epsilon is for an inductor, right? It is self inductance L, which is a value times delta I over delta t. Okay, L is the value of the inductor and it's measured in henries, so you might have a millihenry inductor. Look what happens. The minus signs cancel out and we get L out in front, we get a delta Q over a delta t, and then we still have a delta I. But this is L times delta Q over delta T is I, and then we have a change in I. So when you do this properly with integration you can figure out what the energy in the inductor is, and it is simply this: one-half L I squared. This is similar to what we were talking about with capacitors, right? Remember the capacitor was one-half CV squared. Similar thing, but now we have one-half L I squared. But if you're generating energy in this inductor, there really must be energy in those B fields themselves, because the B field in the center of the solenoid which is really important for the inductor, and so the energy in the B field itself it is the following. And we're going to write down the energy density which is energy per volume, and it is one over two mu naught B squared. Remember, for the electric field we had one-half with an epsilon naught times e squared, now we've got one over two mu naught times B square.