Q2. (25 points) A long cylindrical cable consists of a...

Q2. (25 points) A long cylindrical cable consists of a conducting cylindrical shell of inner radius a and outer radius b. The current denisty J? in the shell is out of the page and varies with radius as J = ?r for a ? r ? b and zero outside of that range, where ? is a positive constant with units A/m³. Find the magnetic flux density in each of the following regions a) r < a, b) a < r < b, and r > b r=b r=a ?? ? ?? Figure 2: Cross section of a long cylindrical cable

Transcript

00:01 So in this question, you are giving a non -cylinder, you know, cable, right, that consists of a conduct cylindrical share, which is basically this is a share, right? this is a share.

00:13 It has inner radius a and out radius b.

00:16 And this current come out of this paper page, right? so the current way it go, come out of the page.

00:23 So it will be, let me just try to indicate it.

00:26 I used to dot to indicate, right? there's a currents everywhere, you know.

00:30 And this current has a density that that is actually proportioned to the radius within this share right now, but zero elsewhere.

00:42 So you ask actually find the magnetic flux density b, right? i call it b in these three regions in this region and this region and also outside this region.

00:54 So let's begin with the inner region.

00:58 Let's look at the inner region.

01:00 First, we should notice that because of the symmetry of this syninder, you know that the magnetic field lines, the magnetic field lines where, you know, everywhere, i mean, you just circle, right? it's just concentric circle.

01:15 So it's going to like this.

01:15 Let me try to indicate these lines.

01:20 So it would be something like, roughly would be something like this, i mean, sure, with a different color.

01:26 It's yellow.

01:28 It would be something like this, right? it would be something like this.

01:32 Circle, right? concentry circles.

01:38 And inside, i also should be like this.

01:40 Oh, inside, let me use a different color.

01:43 And let's use this color.

01:47 So inside it would be something like this, if there's any one.

01:50 And then also outside.

01:53 Let me try to do with this here.

01:56 So the magnet field lines must go round, must circulate in circles, right? i run the same thing.

02:04 Because of the symmetry.

02:06 Now you can use the amphosal, actually define the magnet flux.

02:11 That's the, that is the b, right? you know that according to ample's law, according to ample's law, you know that if you do a circle, for example, if you look at a circle like this, on this line, and if you integrate b along that a circle, along that circle.

02:40 And this must be equal to the current enclosed in this circle...

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