- #1
Aaron121
- 15
- 1
Purcell says that taking the surface integral of the magnetic field ##\textbf{B}## over the surfaces ##S_{1}, S_{2}, S_{3},...## below is a good way of finding the average of the volume integral of ##\textbf{B}## in the neighborhood of these surfaces.
More specifically, he says in page ##553## of the third edition,
More specifically, he says in page ##553## of the third edition,
I can't really see the connection between the volume average of ##\textbf{B}## over, say, the volume between ##S_2## and ##S_3 ## $$\frac{1}{V}\int \textbf{B} dv$$ and the surface integrals of ##\textbf{B}## over ##S_2## and ##S_3## $$\int_{S_3} \textbf{B}\cdot d\textbf{s}_{3},~~\int_{S_2} \textbf{B} \cdot d\textbf{s}_{2}.$$ Any indications?Now taking the surface integral over a series of equally spaced planes like that is a perfectly good way to compute the volume average of the field ##\textbf{B}## in that neighborhood, for it samples all volume elements impartially.