 So to understand electromagnetic induction quantitatively, we have to understand magnetic flux. And flux is not a particularly alien concept. You're worrying about flux whenever you're trying to put your solar panels on your roof so they catch the most sunlight. And you're worrying about flux whenever you're trying to avoid getting wet in the rain. In general, flux is the flow of some physical quantity. So one example might be rain. Rain comes down, comes down in a particular direction, and it comes down sometimes different directions in different places and also different amounts in different places. So you might have a bit where it's raining heavily and hard and going in that direction and then a bit where it's raining quite lightly and more straight down. And so rain is described as a vector field. Which, remember, is just a vector at each point in space. And what are the units of this vector? Well, it's a certain amount of water, which I guess we'd measure in kilograms, the SI unit, per second, per square meter. So if we have a football field, we'll gather more water than if we have a small glass. And so this vector field does units of kilograms of water per second per square meter. So if we wait for an hour, we'll get more water than if we wait for one second. So the other example we talked about was sunlight. Now, sunlight comes in pretty uniformly at a particular angle. If we're out in space looking at a really big scale, it would of course be spreading out in all directions. But down here on Earth, it comes in pretty evenly. So that's also a vector field. And what are the units of that? Well, sunlight gives us energy. So we have a certain amount of energy. And again, we have a certain amount of energy per second and per square meter. Which is also the same thing as a watt per square meter. So you might be trying to catch all this rain into a rain tank so that you can avoid the horrible drought coming up. Or you might be trying to catch this sunlight on a solar panel so you can free yourself from energy costs. Or you might be trying to find out how much magnetic field is going through a loop of wire. Or possibly even multiple loops of wire. So the total flux going through each of these three areas is going to depend on a few things. It's going to depend on the strength of the vector field. So if you have more watts per square meter, you're going to get more energy coming through. If you have more kilograms per second per square meter, you're going to have more water coming through. And if you have more magnetic field, you're going to have more magnetic flux coming through. Secondly, it's going to depend on the size of the area. The bigger the area, the more flux you're going to have coming through. And finally, it's also going to depend on the angle. So it doesn't matter if you have an absolutely enormous solar panel. If you angle it wrong and the sunlight misses it, it goes straight past, it just hits the edge. So it's not exactly the size of the area that matters. It's the size of the area perpendicular to the vector field. So it's really this area here that matters, not the area of the actual solar cell. So the magnetic flux through the coil is usually defined by the symbol capital phi. And it's just equal to the magnetic field times the perpendicular area. Now, if the magnetic field changes direction through your coil, then you've got a more complicated thing to do because you have to break your area up into little pieces of area such that you've got a piece of area so small that the magnetic field is basically going in a constant direction. And then you can calculate the flux through that little piece of area and then add up all those pieces of flux. But we won't do things that are that complicated in calculation. So coming back to electromagnetism, remember that what Faraday found was that current was generated by a changing magnetic field. And after more detailed experiments and analysis, it was found that the electromagnetic force, in other words, the voltage around that loop was equal to the rate of change of the magnetic flux. And so we used the capital delta to describe a change as before. And we used the B to denote the magnetic flux. And so you've got this delta phi over delta t. That's the rate of change of magnetic flux. And so that voltage will be applied to every loop that we have in this coil. And so if we have multiple windings of that coil, we're going to get that voltage for each one of them. And so the total voltage is going to be just the number of loops times the rate of change of magnetic flux. And that is now known as Faraday's law.