Well, I certainly see no math error in t response that I can spot, but I admit I'm clueless about the first part of the equation for diffusion or how it applies to this issue. I need to see it used it a paper if you expect me to comment on it.
You surprised no one.
Michael Mozina needs to retract the following accusation:
I'm not missing and sort of quantification, YOUR side keeps ignoring all the pretty maths.
Here is the equation for magnetic diffusion as stated by
tusenfem:
[latex]
(\sigma \mu)^{-1} \nabla^2 {\bf B} = - \frac{\partial {\bf B}}{\partial t}
[/latex]
Here's the multiple choice question
Michael Mozina couldn't answer:
Here's a simple test to see who's been ignoring the math here: What does the ∇2 mean in tusenfem's first equation?
If that's too hard for you, we can turn it into a multiple choice question:
A. It stands for the Lagrangian operator.
B. It stands for the Laplacian operator.
C. It stands for the curl of a curl.
D. It stands for the gradient of a gradient.
E. It stands for the divergence of a gradient.
Here's a derivation of the equation from Maxwell's equations and Ohm's Law:
[latex]
\begin{center}
\begin{align}
\nabla \times \hbox{{\bf B}} &= \mu \hbox{{\bf J}}
+ \mu \epsilon \frac{\partial \hbox{{\bf E}}}{\partial t} \\
&\approx \mu \hbox{{\bf J}} \\
\nabla \times \nabla \times \hbox{{\bf B}} &\approx
\nabla \times (\mu \hbox{{\bf J}}) \\
&= \nabla \times (\mu \sigma \hbox{{\bf E}}) \\
&= \mu \sigma (\nabla \times \hbox{{\bf E}}) \\
&= \mu \sigma (- \frac{\partial \hbox{{\bf B}}}{\partial t}) \\
\nabla \times \nabla \times \hbox{{\bf B}} &\approx
\mu \sigma (- \frac{\partial \hbox{{\bf B}}}{\partial t}) \\
(\mu \sigma)^{-1} (\nabla \times \nabla \times \hbox{{\bf B}}) &\approx
- \frac{\partial \hbox{{\bf B}}}{\partial t}
\end{align}
\end{center}
[/latex]
where
- (1) is Ampère's law with Maxwell's correction, which is one of the four equations commonly known as Maxwell's equations
- (2) drops Maxwell's correction, which is equivalent to assuming the electric field isn't changing very fast
- (3) takes the curl of both sides
- (4) uses Ohm's Law: J=σE
- (5) moves the scalar constants outside the curl
- (6) uses the Maxwell-Faraday equation, which is another of Maxwell's equations
- (7) restates (3) through (6)
- (8) divides both sides by the scalar constants
Equation (8) above is the same as
tusenfem's equation, except
tusenfem wrote ∇
2 for the curl of a curl. That's weird, confusing, and probably not what
tusenfem intended. I think
tusenfem intended to use the following vector identity:
[latex]
\begin{center}
\begin{align}
\nabla \times \nabla \times \hbox{{\bf B}} &=
\nabla (\nabla \cdot \hbox{{\bf B}}) - \nabla^2 \hbox{{\bf B}} \\
&= \nabla (0) - \nabla^2 \hbox{{\bf B}} \\
&= - \nabla^2 \hbox{{\bf B}} \\
(\mu \sigma)^{-1} (- \nabla^2 \hbox{{\bf B}}) &\approx
- \frac{\partial \hbox{{\bf B}}}{\partial t} \\
(\mu \sigma)^{-1} \nabla^2 \hbox{{\bf B}} &\approx
\frac{\partial \hbox{{\bf B}}}{\partial t}
\end{align}
\end{center}
[/latex]
where
- (1) holds for any vector field
- (2) follows from Gauss's law for magnetism, which is one of Maxwell's equations
- (3) holds because the gradient of a constant field is zero
- (4) uses (1) through (3) to replace the curl of the curl of B in the equation for magnetic diffusion with the negative of the vector Laplacian of B
- (5) multiplies both sides by -1
I think that last equation is what
tusenfem intended to write, and that the minus sign on the right hand side of his equation was just a typo.
So which answer was correct?
Perpetual Student said:
Laplacian of a scalar field is defined as the divergence of the gradient
That can't be what
tusenfem meant, because his equation applies the operator to a vector field.
If the minus sign on the right hand side of
tusenfem's equation was a typo, then ∇
2 is the
vector Laplacian.
While I was typing the above,
tusenfem wrote this:
tusenfem said:
No there is no typo in my equation.
If there was no typo in
tusenfem's equation, then he was using ∇
2 to mean the curl of a curl, which is a confusing notation with which I am not familiar. I await
tusenfem's explanation.