A very large sheet of a conductor located in the x y -plane,...

A very large sheet of a conductor located in the $x y$ -plane, as shown in the figure, has a uniform current flowing in the $y$ -direction. The current density is $1.5 \mathrm{~A} / \mathrm{cm}$. Use Ampere's Law to calculate the direction and the magnitude of the magnetic field just above the center of the sheet (not close to any edges).

Key Concepts

Ampere's Law

Ampere’s Law is a fundamental principle in magnetostatics that relates the integrated magnetic field around a closed loop to the total current enclosed by that loop. It is mathematically expressed as ?B·dl = ??I_enc and is used to determine magnetic fields in situations with high symmetry, such as infinite sheets, wires, or solenoids.

Uniform Current Density

Uniform current density refers to a constant amount of current per unit area flowing through a conductor. This concept is important when analyzing current distributions over surfaces, like a large current-carrying sheet, and simplifies calculations of the enclosed current which is integral for applying Ampere's Law.

Magnetic Field due to an Infinite Current Sheet

For an infinite current-carrying sheet with uniform current density, the magnetic field above and below the sheet is uniform and can be derived using Ampere's Law. The resulting magnetic field is independent of the distance from the sheet (when far from the edges) and its magnitude is typically given by B = (??K)/2, where K is the surface current density.

Right-Hand Rule

The right-hand rule is a mnemonic used in electromagnetism to determine the direction of the magnetic field relative to the direction of the current. For a current flowing in a particular direction, curling the fingers of the right hand around the current-carrying conductor with the thumb pointing along the direction of the current gives the direction of the circulating magnetic field lines.

Transcript

00:01 Hi there, so for this problem, we are told that a very large sheet of a conductor is located in the xy plane, as is shown in the figure, and it has an uniform current flowing in the y direction, and the density is given, and that is equal to 1 .5 ampers per centimeter.

00:32 So we need to use umpers law to calculate the direction in the magnitude of the magnetic field just above the center of the sheet.

00:45 So to find the magnetic field above the center of the surface of a current current sheet, we use umpers law.

00:57 So the path taken should be far from the edges and should be rectangular as shown in.

01:06 In this diagram that we are going to choose.

01:11 So we have some in here, some magnetic field in here.

01:17 So are you going to choose the ones in the middle? we're going to choose this one in here.

01:30 And we will have some path in here.

01:46 So we have the magnetic field b1, the magnetic field b2, so the direction of the magnetic field is found using the right -hand rule to be in the adds positive direction about the surface of the conductor.

02:10 So on per's law states that the close integral of the product between the magnetic field and the bed door of length is equal to the magnetic field times two times pi times the radius is equal to new sub zero times the current enclose so as you can see we're gonna label this we're gonna set that this is one two three and four so we can know that exceptions 1 and 3 are perpendicular to the films.

03:01 So from that we have that the magnetic field times the ds is going to be equal to 0 because those are perpendicular.

03:14 As you can see, the magnetic field is in the paper...

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