A very large sheet of conducting material located in the x y -plane, as shown in the figure, has

A very large sheet of conducting material located in the x y...

Key Concepts

Ampere's Law

Ampere's Law states that the line integral of the magnetic field around any closed loop is directly proportional to the net current enclosed by that loop. This law is especially useful in cases with high symmetry, as it simplifies the calculation of the magnetic field by allowing one to choose an Amperian loop that makes the integral manageable.

Magnetic Field of an Infinite Sheet of Current

An infinite sheet of current creates a magnetic field that is uniform in magnitude on either side of the sheet and whose direction is determined by the current flow. By applying Ampere's Law to such a configuration, one finds that the magnetic field just above or below the sheet is constant and proportional to the current density, assuming an idealized infinite sheet.

Right-Hand Rule for Magnetic Fields

The right-hand rule is a guideline for determining the direction of the magnetic field induced by a current-carrying conductor. When the thumb of the right hand points in the direction of the conventional current, the curl of the fingers shows the direction in which the magnetic field loops, which is essential for correctly identifying the field's orientation relative to the current.

Current Density

Current density is defined as the amount of electric current flowing per unit area of a cross-section. It is a crucial concept for describing how current is distributed in a material, and it plays a key role in the analysis of electromagnetic fields, particularly when applying Ampere's Law to continuous current distributions.

Transcript

00:01 Here we're going to determine the magnetic field above a very large flat current sheet.

00:10 And the way we're going to describe that sheet of current is with a current density.

00:16 Sometimes it's called k to distinguish between i and j, but it is the amount of current per unit length.

00:26 So one way you can think about that sheet is it is made up.

00:30 Of a flat ribbon of wires, for example.

00:34 And that's kind of what i'm showing there, is that the wires are carrying, let's see, they're carrying current into the page, into the y direction, which is what the x means.

00:49 And yeah, i'll just kind of show that what you're thinking of is that there's a whole bunch of wires lined up side by side.

01:00 Now, this is supposed to be a, fairly, you can think about it as an infinite sheet.

01:06 And so we're going to go ahead and use amper's law.

01:12 So the idea with amperer's law is that it can be used if you have a high degree of symmetry, which usually means something very long or infinite.

01:24 And so we'll have to develop an amperian loop that's closed, and we'll have to determine the amount of enclosed current inside of there, and along that loop we will expect the magnetic field to be constant.

01:45 So the loop that will draw, and this may not be obvious, but if we use the right -hand rule, that may be a place to start, is to think about one of those wires.

01:56 So let me just kind of blow up one of those wires carrying a current, one of those imaginary wires, carrying a current into the page, what we would expect the magnetic field to look like is it would circulate around the wire in a circle.

02:19 Okay, and it would be equal strength, equal distance from that wire.

02:25 So a single wire, you have a magnetic field, constant, and a certain distance from the wire.

02:37 Now, because these wires are side by side, what they will do is the side -by -side nature of them will allow cancellation of the magnetic fields that are side -by -side, so to speak.

03:06 So there'll be cancellation off to the side.

03:09 And so we would expect then that a rectangular loop, and i'll try to draw this, and of course you'll have to imagine the rest of the sheet, but a rectangular loop that extends equal distance above and beneath the wires by the same distance.

03:34 I'll call that d, d.

03:41 That that should be our empyrian loop along which we want to calculate the enclosed current.

03:54 All right, so let's take a look.

04:00 We expect that the magnetic field will be constant along that loop, and we expect it to be zero at the endpoints.

04:10 As long as the sheet goes on infinitely, there won't be any contribution from the sides.

04:19 So this contribution is zero, if infinite.

04:23 And remember that there is no such a thing, of course, as an infinite sheet...

Please give Ace some feedback