mmh A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charge

step 1 So this problem we have here is mainly conceptual. So what we have is we have a sheet. So let's

Mmh A very long, hollow cylinder is formed by rolling up a...

Key Concepts

Ampere's Law

Ampere's Law states that the line integral of the magnetic field around any closed loop is equal to ?? times the net current enclosed by the loop. This fundamental law of magnetostatics underpins the calculation of magnetic fields in current-carrying conductors and is essential for determining the field in symmetric configurations.

Symmetry in Magnetic Field Problems

Exploiting symmetry, such as cylindrical symmetry in a long, hollow cylinder, simplifies the calculation of magnetic fields. The magnetic field must be tangential and constant in magnitude along circular loops centered on the axis, allowing one to choose convenient Amperian loops that respect this symmetry.

Amperian Loop and Path Choice

Selecting an appropriate Amperian loop, typically a circle perpendicular to the axis of symmetry, allows for an easier application of Ampere's Law. This loop helps in integrating the magnetic field effectively by taking advantage of constant field magnitude along the path, thereby linking the geometry and the enclosed current.

Enclosed Current

The concept of enclosed current differentiates between regions inside and outside the current-carrying conductor. For points outside the hollow cylinder, the entire current is enclosed, whereas for points inside the hollow region, no current is enclosed, which leads to a magnetic field of zero within the hollow interior.

Transcript

00:02 So this problem we have here is mainly conceptual.

00:06 So what we have is we have a sheet.

00:11 So let's draw a sheet first.

00:19 So it's just like a plane.

00:21 We'll make it a little bit longer so we can see what's happening.

00:26 So we have this sheet.

00:27 It doesn't have to stop here.

00:29 We can say it continues on to infinity.

00:31 But i can't draw to infinity.

00:33 So we'll represent it like that.

00:34 And we take this sheet and we pick it up and we curl it.

00:41 So what we end up happening, so we take this point and touch it to this one, we'll end up with a big cylinder of radius r.

01:04 So that's curling up this sheet.

01:07 Now we're told, so this is a cylinder now and we have this axis.

01:12 Draw that red, we have this axis of the cylinder.

01:17 That goes down the middle of the cylinder.

01:44 So this is a long sheet, and we're told that we have magnetic field, a magnetic field here running parallel to the axis.

01:56 So that means that if our magnetic field is running in either direction here, so it would be in or out of the page, our magnetic field will be in a circle around this wire.

02:18 So we have our magnetic field that travels all the way around.

02:21 Completes a circle.

02:30 Now we want to show or we want to find out what the magnetic field looks like outside of this cylinder.

02:38 So we have amper's law that states that the sum of the magnetic fields that are parallel to the current, or not the current, sorry, parallel to a small change in length is equal to mu not the permeability of free space times the current so once again we have to kind of figure out what the left hand side of this equation is telling us so we have our d .l if we take a small portion of our cylinder here and cut it into a dl so this is our delta l if we were to take this delta to l and make it really, really small, it will appear as if it's straight, as if it's a straight line, kind of like the surface of the earth.

03:49 We're so close to it that it looks like it's flat.

03:53 So it looks, if we were to take this and shrink it down really, really small, it looks like we have a straight line here.

03:59 In that case, the magnetic field will point, depending on which way our current is flowing either to the right or to the left.

04:07 So it is parallel.

04:09 At every point, if we split this up entirely all the way around into the same length dl, at every point the magnetic field is going to be parallel.

04:20 Which means there's some b parallel turns into just b.

04:25 So we have this whole thing here.

04:31 Just turns into the magnetic field.

04:33 B.

04:34 We don't have to worry about any summation.

04:35 It's still the sum of the delta l's though.

04:38 So this, so now this part, becomes a sum because this sum encompasses everything on the left -hand side of the equation.

04:48 This is the sum of the change in l, the delta l, and that's still equal to the same thing on the right -hand side.

05:01 So we found out part of our left -hand side.

05:05 So we need to figure out this part now.

05:09 So what's the sum if we break this entire radius up into little sums of delta -l? we add them all together, it gives us the entire length here, which is the length of our rectangle that we had here.

05:26 So it'll give you this length.

05:28 And we can calculate that with the radius.

05:30 That's just equal to the circumference.

05:33 So we have the sum of delta l is equal to the circumference of our outside of the cylinder, which is equal to 2 pi times the radius.

05:56 That's a terrible pi.

05:58 Let's try that again.

06:00 2 pi times the radius...

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