Hey, guys. In this video, we're going to be talking about Ampere's law and how to use it. Alright? Let's get to it. Any magnetic field, b, must satisfy the following equation. This is what's known as a line integral. Okay? I am integrating b along some line that creates an arbitrary shape, which I called s. Okay? And this line integral always equals μ₀ I_enclosed. Okay? Notice that this integral has this little circle in the integral sign. All that means is that it's a closed line. Okay? We saw this before for Gauss's law, where we use the circle for the surface integral. Okay? And we said that that was a closed surface. So this has to be a closed loop or a closed line, which we call an amperean loop. Okay? And what Ampere's law says is that this integral depends only on the current enclosed by that amperean loop. And a consequence of that is that the magnetic field is only going to depend upon the current enclosed by that amperean loop as well, just like we saw in Gauss's law. Okay? How to apply this is pretty straightforward. If we take this arbitrary curve s in the figure above me, which surrounds a current I, everywhere along this loop, I can measure the magnetic field, and I can do the dot product with dl. All dl is is it's a very small it's an infinitesimal segment that goes along the curve. Right? So at this point, dl is going to be like this. This point, dl is going to be like this. It just points tangential to the curve. Right? Tangential everywhere. And the magnetic field is going to depend it's going to point in something like this. It would look something like this. It would look something like this. Something like this. Something like this, etcetera. And I could do the dot products, add all those up, integrate it along the curve s, and that would equate to a measurement times the charge enclosed. Okay? Let's apply Ampere's law for a scenario that we've already seen a bunch of times. Using Ampere's law, find the magnetic field due to an infinitely long current carrying wire. So I'm just going to draw the current as going into the page. Okay. Exactly like the figure above. Now, what does that magnetic field look like? Alright. Well, I'm going to stick my thumb in the direction of that current into the page, and the magnetic field is going to curl around it exactly like that. Okay? Which means if I were to draw an empyrean loop that was a circle of some radius r, the magnetic field is always going to be parallel to that line segment dl. Remember, that line segment always points tangential to the curve. Right? So Ampere's law says that this integral, b⋅dl, is going to just be bdl. That dot product is always going to be 1 because they always point parallel. And this equals μ₀ I_enclosed. Okay? Now just like we did for Gauss's law, we cheated a little bit because we knew what the magnetic field looked like. We had to cheat with Gauss's law as well by knowing what the electric field looks like. Furthermore, along this circle, because it's of a constant radius, the magnetic field is constant. So as I go along this circle, as I change my dl, the magnetic field doesn't change. So I can pull that field out. Okay? And lastly, I just need to solve this integral. Well, if I go around the circle one revolution, I've covered a distance equal to the circumference. So that magnetic field is just 2 pi r, and I enclosed is just I. I over 2 pi r. Exactly what we would expect it to be, and what we've seen it to be over and over again. Okay? So Ampere's law tells us what the magnetic field due to an infinitely long wire is much, much more quickly than Biot Savart law does, for instance. Alright. Thanks for watching, guys.