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Physics

Learn the toughest concepts covered in Physics with step-by-step video tutorials and practice problems by world-class tutors

29. Sources of Magnetic Field

Ampere's Law (Calculus)

1
concept

Ampere's Law with Calculus

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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.
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example

Magnetic Field Inside a Solenoid

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Hey, guys, let's do an example. What is the magnetic field along the axis of a solenoid? So let's take some sort of soul annoyed like this, Okay? It's got some current going through it. It's got in a number of loops spread out over a distance. Yeah. Okay. Now, what is the magnetic fields gonna look like straight down the center of this solenoid? Okay. We want to use NPR's law to find what that magnetic field is. Now, NPR's law tells us that the line integral B D L across an imperial loop has to equal mu, not times the charge enclosed. Now the particular and period loop I'm gonna choose. Looks like this. I'm going to go down the axis off the soul, annoyed on Lee for the length of the solenoid where I know that the magnetic field is constant and straight like for Gazans. Law, we're gonna cheat a little bit because we already know things about what the magnetic field should be before using amperes law. Then I'm gonna go straight up the 90 degree angle straight up infinitely high. So there's a gap here, and then I'm gonna come back across, okay, This little gap here takes us all the way up to infinity. Okay, so this imperial loop has four steps. It has Step one, which is along the axis Step two, which is straight up perpendicular to the axis all the way up to infinity Step three, which is parallel to the axis coming back but infinitely far away. And Step four, which is perpendicular to the axis coming back down. So this integral becomes the first path b dot e l plus the second half b dot de l plus the third path b dot de l plus the fourth path v dot pl Now the thing about paths to and pats four is that above the soul annoyed the magnetic field line points parallel to the axis. So you're gonna get a magnetic field line that is perpendicular to D l Here's d l. Well, that magnetic field line actually points in the opposite direction. Okay, because they're perpendicular. The dot product is always zero So right away for two. And for 40 Now, what about three? The thing is, three is parallel to the axis. So there would be a component of the magnetic field along it The problem is that it's infinitely far away. Things in physics always dropped to zero when you go infinitely far away. Otherwise, to objects that are infinitely far apart could still interact with one another. Gravity goes to zero Gravitational potential energy goes to zero Electric force, electric potential energy, electric potential electric field, etcetera They all become zero infinitely far away so that two things can't interact when they're infinitely far away. So the magnetic field line the long path This only leaves path one okay, and they're parallel along path one. So this integral becomes B D. L from zero toe l just the length of that axis which runs the length of the soul annoyed capital l Okay, the magnetic field is gonna be constant along the axis so I can pull it out. And this just becomes B L. Now, this is the left half of amperes law. What about the right half of amperes law that tells us its knew, not times the enclosed current. If there was a single loop here, just a single loop, we would have a single current I For each loop, we have an additional current I right how many loops air there there are in loops. So this is times in each of them carries I Okay, so if I got the way just so we see the equation right here, this whole thing becomes bl he goes mu not in I or B is mu not capital in over. L I This is the exact equation for a solo that we're expecting sometimes written like this where little in is the number of terms per unit life. Okay, guys, Thanks for watching.
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Problem

A solid, cylindrical conductor carries a uniform current density, J. If the radius of the cylindrical conductor is R, what is the magnetic field at a distance ? from the center of the conductor when r < R? What about when r > R?

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