No Magnetic Fields: Divergence & Curl of B=0

In summary, the conversation discusses the conditions for a field to be conservative and the implications of setting B equal to the curl of A, which is defined as the gradient of a scalar function. However, this is incompatible with Ampere's law, as the curl of a gradient is always zero. Therefore, the assumption that A is a gradient of some scalar is not justified. Additionally, the conversation clarifies that the integral of the rotor over a surface of a volume always gives zero, as it can be split into two parts with a curve and the integrals cancel each other out. This highlights the difference between an integral over a closed surface and the integral around a closed loop, which defines a conservative field.
  • #1
jackxx
1
0
[tex]\nabla \cdot B=0,[/tex]
so [tex]\int\nabla \cdot B dv=0,[/tex]
then [tex]\int B \cdot \widehat{n}da=0,[/tex]
let [tex]B=\nabla \times A,[/tex]
then [tex]\int\nabla \times A \cdot \widehat{n}da=0,[/tex]
thus [tex]A=\nabla\varphi[/itex], thus [tex]B=\nabla \times A=\nabla \times\nabla\varphi=0[/tex]

I know something is wrong with that but I am not sure what it is any ideas?
 
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  • #2
The problem per se arises when you define A as a gradient of some scalar. This would be incompatible with Ampere's law mainly because the curl of a gradient is always zero. So in this case you are specifically setting B = 0 from the start essentially.
 
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  • #3
A field is conservative if it's integral over any closed curve is zero. This condition is satisfied if the integral of field's rotor over area limited by the curve is zero. In your case you choose whole surface (of a volume) for the area: the surface is not an area limited by a curve, so you can't use the fact that integral of rotor over the surface implies that integral of field over a curve is zero. In fact the integral of rotor over a surface of a volume always give zero (for any vector field), because you can split the surface on two parts with a curve and the integrals of rotor over two parts of surface give +/- integral of field over curve, so they cancel each other out.
So the assumption that A=grad(fi) in not justified.
 
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  • #4
You just confused an integral over a closed surface with the integral around a closed loop defining a conservative field.
We are lucky you are wrong. Otherwise all electric motors would instantly stop.
 

Related to No Magnetic Fields: Divergence & Curl of B=0

1. What is meant by "No Magnetic Fields: Divergence & Curl of B=0"?

"No Magnetic Fields: Divergence & Curl of B=0" refers to a condition in which the magnetic field vector, represented by the symbol B, has a magnitude of 0 at all points in space. This means that there is no magnetic field present in the given region or system. The terms "divergence" and "curl" refer to mathematical operations used to describe the behavior of the magnetic field.

2. Why is it important to study "No Magnetic Fields: Divergence & Curl of B=0"?

Understanding the behavior of magnetic fields with no divergence or curl is crucial in many areas of science and technology. This condition can occur in certain physical systems, and studying it can provide insights into the behavior of electromagnetic phenomena and help engineers design more efficient devices.

3. What are the implications of "No Magnetic Fields: Divergence & Curl of B=0" in terms of Maxwell's Equations?

In Maxwell's Equations, which describe the fundamental laws of electromagnetism, the absence of magnetic fields with no divergence or curl is represented by the equation ∇·B = 0 and ∇×B = 0. These conditions can be used to solve for other physical quantities, such as electric fields and currents, and are an important part of understanding electromagnetic phenomena.

4. How is "No Magnetic Fields: Divergence & Curl of B=0" related to the concept of magnetic monopoles?

Magnetic monopoles are hypothetical particles that have only one magnetic pole, either a north or south pole, instead of both. The absence of magnetic fields with no divergence or curl is related to the non-existence of magnetic monopoles in our universe. If magnetic monopoles did exist, it would result in a non-zero divergence and curl of the magnetic field.

5. What experimental evidence supports the existence of "No Magnetic Fields: Divergence & Curl of B=0"?

One of the main pieces of evidence for the existence of this condition is the observation of superconductors, which are materials that have zero electrical resistance when cooled below a certain temperature. In superconductors, the magnetic field is expelled from the material, resulting in a region with no magnetic field and satisfying the conditions of no divergence and curl. Additionally, studies of certain physical systems, such as plasmas and tokamaks, have shown that the magnetic field can be controlled to have no divergence or curl in certain regions.

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