Magnetic field of an infinite layer

[Dell]

given a n infinite layer, with a thickness of D and a current density of J direction in the diagram (on x-), prove that the magnetic field above the layer is constant irrespective of the height 'z'

http://lh4.ggpht.com/_H4Iz7SmBrbk/Si1JMFGiWEI/AAAAAAAABEE/LVo64TBgsmw/Untitled.jpg

i had a similar problem in electrostatics and what i did was use gauss law, here what i want to do is use amperes law, the problew is finding the correct path to use. i think that it must be a rectangle path therefore my closed integration will be in 4 parts, 2 along the y axis, 2 along the z axis,, i thought of taking a path of length L, at a height Z,

somehow i need to get 0 for the integration of the heights, so that my equation is independant of z

 

[Andrew Mason]

given a n infinite layer, with a thickness of D and a current density of J direction in the diagram (on x-), prove that the magnetic field above the layer is constant irrespective of the height 'z'

http://lh4.ggpht.com/_H4Iz7SmBrbk/Si1JMFGiWEI/AAAAAAAABEE/LVo64TBgsmw/Untitled.jpg

i had a similar problem in electrostatics and what i did was use gauss law, here what i want to do is use amperes law, the problew is finding the correct path to use. i think that it must be a rectangle path therefore my closed integration will be in 4 parts, 2 along the y axis, 2 along the z axis,, i thought of taking a path of length L, at a height Z,

somehow i need to get 0 for the integration of the heights, so that my equation is independant of z

Take any rectangular path enclosing the full thickness of a section of the slab that is in the yz plane.

By symmetry, what can you say about the component of B in the z direction? What does this say about $\int B\cdot ds$ over the two sides of the rectangle in the z direction?

AM

 

[nlightner]

I would approach this problem using the principles of electromagnetism and mathematical analysis to prove that the magnetic field above the infinite layer is constant regardless of height.

Firstly, we can use the Biot-Savart law to determine the magnetic field at any point above the infinite layer. This law states that the magnetic field at a point is directly proportional to the current density and inversely proportional to the distance from the current element. Since the current density is constant and the distance from the current element (the infinite layer) is also constant, we can conclude that the magnetic field will be constant at any point above the layer.

Next, we can use the principle of superposition to determine the total magnetic field at a point above the infinite layer. This principle states that the total field at a point is the vector sum of the individual fields produced by each current element. Since the layer is infinite, we can assume that there are an infinite number of current elements contributing to the total field. However, since the current density is constant and the distance from each current element is constant, the individual fields will all be in the same direction and will therefore add up to give a constant magnetic field.

Now, to prove that the magnetic field is independent of height, we can use Ampere's law. This law relates the magnetic field to the current enclosed by a closed path. In this case, we can take a rectangular path of length L and width D, enclosing the infinite layer. The current enclosed by this path will be J multiplied by the area of the layer, which is D. Therefore, the current enclosed is JD.

Now, according to Ampere's law, the closed integral of the magnetic field around this path is equal to the current enclosed multiplied by the permeability of free space (μ0). Mathematically, this can be written as:

∮B·dl = μ0I

Since we have already established that the magnetic field is constant and the current enclosed is also constant, we can conclude that the closed integral of the magnetic field is also constant. This means that the magnetic field above the layer will be constant regardless of the height above the layer.

In conclusion, using the principles of electromagnetism and mathematical analysis, we can prove that the magnetic field above an infinite layer with constant current density is constant and independent of height. This has important implications in the study of electromagnetism and can be applied to various real-world situations where an infinite layer