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Biot-Savart Law with Calculus

Patrick Ford
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Hey, guys, in this video, we're going to be discussing the general calculus form of the videos of our PLA. All right, let's get to it for any current okay? Contained by some wire. If we want to measure the magnetic field at some position, our vector away. Okay, so we have this figure here above me where we have some arbitrary wire in this case, a curved one with some arbitrary current. I we want to measure the magnetic field here. Okay. What's that going to look like? We'll be use of our law tells us that this is mu not over four pi times that integral of the currents D l cross our vector over our cubes. Okay. We already know what are is the only thing we need to know is what d l is, okay. And D l is a very, very small vector, okay? In the direction of the current. Wherever this current is pointing, I can choose a really tiny, infinitesimally small vector D l in that direction. Okay. And the cross product right here is between that vector D l and the position vector are okay. BeOS of art law. This integral equation reduces to our familiar magnetic field equations. For instance, we have the magnetic field due to a point charge, and we have the magnetic field due to an infant long current carrying wire as just two examples of what the bills of our law reduces to. Okay, let's do an example. Show that the abuse of our law for occurring is the same as the equation above for a point charge. So if instead of having just some large current which is made up of a bunch of streaming electrons, we have a single charge, Q That's moving. And we wanna use bills of our law given to us the general form to find it for a point charge. Okay, the general form is just Muna over four pi integral of I d l cross our vector over are cute. Okay, What we're gonna do here is we're gonna substitute in the definition for current current by definition, is de que DT Okay, so I'll plug that in. This becomes mu not over four pi integral dick You d t d l cross our over are cute. Now there's something that you can do what you learned in calculus called implicit differentiation. Okay, if you treat these infinite testicles D Q d t d l as implicit differentials. Okay, we can reorder them. And we can just say that this is mu not over four pi integral de que d l DT Cross our vector over our cute. Okay, now what d l is is it's a very, very small amount of distance in the direction that the current is moving or in this case, in the direction of the charges moving the rate at which that's increasing in size is just the velocity of the charge. So it becomes you not over for pie. Integral de que the cross our over our cute. Now what we're integrating over is our charge, Dick. There is a single charge here, so the velocity doesn't change for the charge, and neither do these positions. Those positions are just the position that we want to measure the magnetic field in so we can pull all of that out. Simplifying the cross product. This becomes VR. Signed Fada over r cubed. Sorry, I'm missing the four pi integral of Dick. And there's just a single charge. The integral dick is just that charge Q So it becomes Muna over four. Pi que by the way we lose the power of our in the denominator Q v sine theta over r squared. And that is our familiar equation for the bios of art law for a single moving charge. All right, Thanks for watching guys.