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.
