Problem 5 Use the Biot-Savart law to find the B-field of a...

Problem 5 Use the Biot-Savart law to find the B-field of a straight segment of wire carrying a current I. (a) Show that the magnitude of the B-field at a distance r from the wire is: $B(r) = \frac{\mu_0 I}{2\pi r} \frac{\sin \theta_1 + \sin \theta_2}{2}$. What is the direction of this B-field? (b) Show that this B-field violates Ampere's law unless the wire is infinitely long.

Transcript

0:00 All right, hello.

00:01 In this question, we're told we have a current carrying wire.

00:04 It's carrying a current of i1 out of the page, so it's an arrowhead coming towards us.

00:09 And in part a, we're asked to use ampere's law to determine the magnetic field b at a distance r, sorry, from the wire.

00:15 So ampere's law says that we have the integral of b dotted with dl equals mu naught times our enclosed current.

00:27 So this right -hand side is innocent enough, but this left -hand side is kind of, what does that all mean? well, that essentially means if i draw an amperian loop, so i draw some nice symmetrical circle or any loop, but in this case, i'm going to do a nice circle with some radius r.

00:44 If i draw an amperian loop, i say that if i sum up all of the b dot dls, so i have a current carrying wire, i'm going to also have a generated b field, which will consequently be pointing in the same direction.

01:02 If i sum up all of those little bits of b dl, which means i'm going to take the integral around this line, i'm going to end up with that value being mu naught times whatever charge carriers i have enclosed.

01:17 In other words, i'm going to end up with a total magnetic field b, and then i just have to sum that up over this whole loop here.

01:26 Well, what is all of, if i add together all of these little dls around this loop, i'm going to end up just with the circumference of a circle, right? so i'm going to have 2 pi times r, and that's going to equal mu naught times i enclosed there.

01:42 I want to solve what b is going to be, so that's going to be mu naught i enclosed over 2 pi r, where r is the radius i am away, and that's my amperian loop.

01:52 What is the direction i'm not going to be? well, to do this, we're going to use the right hand rule.

01:56 So if i take my right hand and point my thumb in the direction of the current, which is out of the page, so the thumb towards my face, and i curl my fingers, that's the direction that the b field will be pointing.

02:08 So if i curl them at the top, i'm going to have a b field going to the left.

02:12 At the bottom, my fingers will curl to the right, so on and so forth.

02:16 So this is going to be, our b field here is going to have this magnitude, and it's going to be going in the counterclockwise direction for part a.

02:27 So simple enough...

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