Why can't we use Ampere's Law?

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Discussion Overview

The discussion revolves around the application of Ampere's Law to determine the magnetic field at a specific point due to a current-carrying sheet. Participants explore the conditions under which Ampere's Law can be effectively applied versus when alternative methods, such as modeling the sheet as a collection of infinitely long wires, are necessary.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests modeling the current sheet as infinitely long wires to find the magnetic field, questioning why Ampere's Law cannot be used directly.
  • Another participant references a previous discussion indicating that Ampere's Law is applicable in this scenario.
  • A participant describes using an Amperian Loop to find the magnetic field inside and above the slab, noting that the magnetic field is perpendicular to the length elements along the height, leading to a zero integral on the left side of Ampere's Law.
  • One participant corrects themselves, acknowledging the finite width of the sheet and stating that the lack of symmetry in the integral path prevents Ampere's Law from providing a solution.
  • Another participant proposes using a circular or rectangular loop that encloses the entire cross-section of the sheet, questioning the symmetry of the magnetic field geometry.
  • A participant clarifies that the geometry is asymmetric rather than anti-symmetric, explaining that this asymmetry results in a non-uniform magnetic field that cannot be simplified in the integral.
  • One participant expresses understanding of the term "asymmetric" in relation to the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Ampere's Law, with some suggesting it can be used while others argue it cannot due to the lack of symmetry. The discussion remains unresolved regarding the best approach to solve the problem.

Contextual Notes

There are limitations regarding the assumptions made about the geometry of the current sheet and the symmetry of the magnetic field, which affect the applicability of Ampere's Law.

Kyle Nemeth
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We are asked to find the magnetic field at point P, all of the quantities in the figure are known values and the current density is uniform. One way to solve this problem is by modeling the sheet as a collection of infinitely long wires, with each wire contributing an amount of magnetic field dB and then integrating to find the total magnetic field. Why is it that this approach must be used and not an approach involving Ampere's Law directly?
 

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I am familiar with this problem, the Amperian Loop is chosen as a rectangle within the slab with a height smaller than the thickness of it (to find B at a point within the slab) and can then be chosen as a rectangle with a height larger than the thickness of the slab (to find B at a point above the slab) and this is okay because the magnetic field is perpendicular to the length elements along the height, so the integral on the left side of Ampere's Law is 0. The plane slab actually extends infinitely in two dimensions rather than just one as in my problem, I apologize I should have specified that the sheet is "thin" so that it has no thickness.
 
My mistake: I see you have a finite width ## w ## to the sheet. You don't have enough symmetry on the integral path for Ampere's law to supply the answer. The ## B ## in the integrand is non-uniform.
 
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Awesome, thank you, I've thought about this also, so if we choose a circle as our loop, or even a rectangle (so that the entire cross-section of the sheet is enclosed) it is true that both would be anti-symmetric with the geometry of the magnetic field formed by the sheet?
 
Kyle Nemeth said:
Awesome, thank you, I've thought about this also, so if we choose a circle as our loop, or even a rectangle (so that the entire cross-section of the sheet is enclosed) it is true that both would be anti-symmetric with the geometry of the magnetic field formed by the sheet?
Not anti-symmetric, but asymmetric. If it were simply anti-symmetric, then ## \oint B \cdot dl=0 ##. When it is asymmetric, ## B ## is non-uniform, and can't be removed from the integral in any fashion.
 
ASYMMETRIC, okay, well understood, thank you for your response.
 
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