My question is similar to this one (Boundary conditions magnetic field) in that it is related to the boundary conditions of the magnetic field (B-field). However, my question focuses on mathematically characterizing physicists' claims regarding the boundary conditions of the B-field (eq2 below). Simply put, my question is the mathematical proof of Conjection, below.
Assume that the space $\mathbb{R}^3$ is subdivided into two connected components $M_1$ and $M_2$ by a smooth 2-dimensional manifold $S$ (the figure enlarges only a part of $S$). That means, $\mathbb{R}^3 =M_1 \cup S \cup M_2$ and $M_1$,$S$,$M_2$ are disjoint.
In the physics, when considering the B-fields in materials, the B-field, $\textbf{B}: \mathbb{R}^3 \to \mathbb{R}^3$ is $C^{\infty}$ at both inside of $M_1$ and inside of $M_2$, but they did not mention of the behavior of the $\textbf{B}$ on the $S$, that is the boundary between $M_1$ and $M_2$.
However, according to Maxwell's equations, the B-fields are characterized as a vector field that satisfies $div[B]=0$. This means that if we take a micro cylinder $V$ that 'sandwiches' the curved surface $S$ in the sense of definition 1 below, $$\int \int_{S_1}\textbf{B} +\int \int_{S2}\textbf{B} +\int \int_{S_3}\textbf{B} =\int \int \int_{V} div[B] =0 \ \ (eq,1)$$
Here,
- the radius of this micro cylinder $V$ is $r$ and the height is $h$.
- ${S}_{1}$ represents the upper base of this micro cylinder
- ${S}_{3}$ represents the side surface of this micro cylinder
- ${S}_{2}$ represnts the bottom surface of this micro cylinder
Perhaps if we take a mathematically accurate position, more precisely, $div[B]=0$ should be rephrased as follows:
- For all $\mathbf{x} \in Int[M_1]\cup Int[M_2]$ , $div[B](x)=0$
Def 1 (Sandwiching $S$ at $\textbf{x}_0\in S$)
Suppose that
- a smooth surface $S$ separates $\mathbb{R}^3$ into two connected components, $M_1$ and $M_2$,
- $\textbf{x}_0\in S$ ,and
- $\mathbf{n}(x_0)$ is the unit normal of $S$ at $\textbf{x}_0\in S $,
Then, the micro cylinder$V$ sandwiching $S$ at $\textbf{x}_0$ , iff all of the following conditions are satisfied;- $\textbf{x}_0\in Int[V]$, Int means open kernel
- $S_1\subset M_1 $ and $S_2\subset M_2$
- $S_1 \perp\mathbf{n}(x_0)\ $ and $S_2\perp\mathbf{n}(x_0)$
- Both of the distance between $S_1$ and $\textbf{x}_0$ and the distance between $S_2$ and $\textbf{x}_0$ are $h/2$.
In other words, in the figure below, in (1) $V$ sandwiches $S$, but in (2) it does not sandwich $S$ in the sense of Definition 1;
So far, the story may be clear to me, but...
Physicists argue that by 'making' $S_2$ sufficiently small compared to $S_1$ and $S_3$, the following holds; $$\lim_{r\to 0, h\to 0} \textbf{B}(\textbf{x}_0+ \frac{h}{2}\mathbf{n}(\textbf{x}_0)) \cdot \mathbf{n}(\textbf{x}_0) =\lim_{r\to 0, h\to 0} \textbf{B}(\textbf{x}_0 - \frac{h}{2}\mathbf{n}(x_0)) \cdot \mathbf{n}(\textbf{x}_0) \ \ (eq2)$$
However, mathematically, there seems to be at least the following gap between (eq1) and (eq2);
- Is it possible to make $S_1$ and $S_2$ become infinitely small while keeping $S_3$ sufficiently small relative to $S_1$ and $S_2$?
- When the side surface,$S_3$ is made sufficiently small compared to the top and botttom surfaces, $S_1$ and $S_3$, can $V$ sandwichs $S$ in the sense of Def1?
I think this question will probably be resolved if the following predictions are correct.
Conjection:
Suppose that a smooth surface $S$ separates $\mathbb{R}^3$ into two connected components $M_1$ and $M_2$.
Let $\textbf{x}_0\in S$, and $\mathbf{n}(x_0)$ is the unit normal of $S$ at $\textbf{x}_0\in S $,At this time, there are
- ${r_i} , h_i \ (i=1,2,\cdots) $, sequence of $\mathbb{R}$ and,
- $V_{i} \ (i=1,2,\cdots) $, a series of micro cylinders with radius $r_i$ and height $h_i$
such that,
- (0)$r_i>0$ and $h_i >0$, for all $i=1,2,\cdots $
- (1) $V_{i}$ sandwiches $S$ at $\textbf{x}_0$ in the sense of Def 1, for all $i=1,2,\cdots$
- (2)$\lim_{i\to \infty} h_{i} = \lim_{i\to \infty} r_{i} = 0$
- (3)$\lim_{i\to \infty} h_{i} /r_{i} = 0$
Of course, it would be relatively easy to create $h_i$ and $r_i$ that satisfy the conditions (2) and (3) above; for example, if the following is defined, these satisfy (2) and (3) above.
- $r_i := 1/i$
- $h_i := {r_i}^2$
However, whether the above point sequence satisfies (1) probably depends on $S$.
My questions:
- Is the above Conjection correct, or can it be corrected by adding some conditions?
- If so, please provide proof (with additional conditions), and
- if it is difficult to correct, please provide a counterexample.
See also
