Proving: \int\textbf{B}\cdot\textbf{H}d^{3}x=0

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Homework Help Overview

The problem involves proving that the integral of the dot product of the magnetic field \(\textbf{B}\) and the magnetic field intensity \(\textbf{H}\) over a volume is equal to zero, specifically in the context of a scenario with no current density.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to use a vector identity and the divergence theorem to relate the integral to a surface integral, referencing Ampere's law. Other participants question the surface of integration and the values of \(\textbf{B}\) and \(\Phi_M\) along that surface. There is a discussion about whether the integral can be argued to be zero based on the behavior of magnetic flux in an enclosed region without magnetization.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and questioning the assumptions made about the magnetic fields involved. Some guidance is provided regarding the nature of the integral and its relation to magnetic flux, but no consensus has been reached.

Contextual Notes

The problem is noted to be intentionally general, with no specific conditions or parameters provided, leading to varied interpretations and approaches among participants.

Old Guy
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Homework Statement


Prove [tex]\int\textbf{B}\cdot\textbf{H}d^{3}x=0[/tex]. There is no current density.



Homework Equations




The Attempt at a Solution

Through a vector identity and the divergence theorem, I get
[tex]\oint\Phi_{M}\textbf{B}\cdot{d}\textbf{a}[/tex] but don't know how to proceed. This seems close to Ampere's law with no enclosed current, but not quite.
 
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What surface are you integrating over? What are the values of [itex]\textbf{B}[/itex] and [itex]\Phi_M[/itex] along that surface?
 
He told us the problem was given to us intentionally very general, so none is specified. Could I argue that for an enclosed region in space with no enclosed magnetization, the integral is zero because all the flux in goes out again (kind of like the EM flux argument)?
 
Old Guy said:
He told us the problem was given to us intentionally very general, so none is specified. Could I argue that for an enclosed region in space with no enclosed magnetization, the integral is zero because all the flux in goes out again (kind of like the EM flux argument)?

No, I don't think that works...

[tex]\oint\Phi_{M}\textbf{B}\cdot{d}\textbf{a}[/tex]

does not represent the magnetic flux.

What is the exact wording on the original question? (If it's a problem from Jackson, just state the problem number)
 

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