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Ampere's Law with Calculus

Patrick Ford
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Hey, guys, in this video, we're gonna be talking about amperes law and how to use it. All right, let's get to it. Any magnetic field B must satisfy the following equation. This is what's known as a line integral. I'm integrating. Be along some line that creates an arbitrary shape, which I called s okay. And this line integral always equals mu, not times 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 KAOS is 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 imperial loop. Okay. And what amperes loss says is that this integral depends on Lee on the current. Enclosed by that by that imperial loop. And a consequence of that is that the magnetic field is on Lee going to depend upon the current enclosed by that imperial loop as well. Just like we saw in God'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 D l l D l is It's a very small It's an infinite decimal segment that goes along the curve. Right. So at this point, deal is gonna be like this. This point deal is gonna be like this. It just points 10 gentle to the curve, right, Tangential everywhere. And the magnetic field is going to depend. It's gonna point in the direction that depends on the current enclosed. But at each point, I confined the magnetic field. Okay, So, for instance, here, the magnetic field 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 integrated along the curve s, and that would equal new, not times the charge in close. Okay, let's apply amperes law for a scenario that we've already seen a bunch of times using amperes law find the magnetic field due to an infinitely long current carrying wire. So I'm just gonna draw the current as going into the page. Okay? Exactly Like the figure above. Now, what does the magnetic field look like? All right, well, I'm gonna stick my thumb in the direction of that current into the page in the magnetic field is going to curl around it exactly like that. Me. Okay, which means if I were to draw an imperial loop, that was a circle of some radius r. The magnetic field is always gonna be parallel to that line. Segment D. L remember that line segment always points tangential to the curve. Right. So amperes law says that this integral b dot de l is gonna just be b d l that dot product is always gonna be one because they always point parallel. And this equals mu not. I enclosed now, just like we did for Gaza's law. We cheated a little bit because we knew what the magnetic field looked like. We had to cheat with God'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 d l The magnetic field doesn't change so I can pull that magnetic 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 two pi r and I enclosed is just i una I over two pi are exactly what we would expect it to be and what we've seen it to be over and over again. Okay, So, amperes law tells us what the magnetic field due to an infant long wire is much, much more quickly than Bill Savar law does. For instance, Alright, thanks for watching guys.