9—Vector Calculus 1 230Magnetic Boundary Conditions
The equations for (time independent) magnetic fields are
∇×B~=μ 0 J~ and ∇.B~= 0 (9.49)
The vectorJ~is the current density, the current per area, defined so that across a tiny areadA~the
current that flows through the area isdI =J~.dA~. (This is precisely parallel to Eq. (9.1) for fluid
flow rate.) In a wire of radiusR, carrying a uniform currentI, the magnitude ofJisI/πR^2. These
equations are sort of the reverse of Eq. (9.36).
If thereisa discontinuity in the current density at a surface such as the edge of a wire, will
there be some sort of corresponding discontinuity in the magnetic field? Use the same type of analysis
as followed Eq. (9.47) for the possible discontinuities in the potential function. Take the surface of
discontinuity of the current density to be thex-yplane,z= 0and write the divergence equation
∂Bx
∂x
+
∂By
∂y
+
∂Bz
∂z
= 0
If there is a discontinuity, it will be in thezvariable. PerhapsBxorBzis discontinuous at thex-yplane.
The divergence equation has a derivative with respect tozonly onBz. If one of the other components
changes abruptly at the surface, this equation causes no problem — nothing special happens in thex
orydirection. IfBzchanges at the surface then the derivative∂Bz/∂zhas a spike. Nothing else in
the equation has a spike, so there’s no way that you can satisfy the equation. Conclusion: The normal
component ofB~ is continuous at the surface.
What does the curl say?ˆx
(
∂Bz
∂y
−
∂By
∂x
)
+ˆy
(
∂Bx
∂z
−
∂Bz
∂y
)
+ˆz
(
∂By
∂x
−
∂Bx
∂y
)
=μ 0
(
ˆxJx+yJˆ y+ˆzJz
)
Derivatives with respect toxorydon’t introduce a problem at the surface, all the action is again
alongz. Only the terms with a∂/∂zwill raise a question. IfBxis discontinuous at the surface, then
its derivative with respect tozwill have a spike in theyˆdirection that has no other term to balance
it. (Jyhas astephere but not a spike.) Similarly forBy. This can’t happen, so the conclusion: The
tangential component ofB~ is continuous at the surface.
What if the surface isn’t a plane. Maybe it is a cylinder or a sphere. In a small enough region,
both of these look like planes. That’s why there is still* a Flat Earth Society.
9.10 Index Notation
In section7.11I introduced the summation convention for repeated indices. I’m now going to go over
it again and emphasize its utility in practical calculations.
When you want to work in a rectangular coordinate system, with basis vectorsˆx,yˆ, andzˆ, it is
convenient to use a more orderly notation for the basis vectors instead of just a sequence of letters of
the alphabet. Instead, call themˆe 1 ,ˆe 2 , andeˆ 3. (More indices if you have more dimensions.) I’ll keep
the assumption that these are orthogonal unit vectors so that
ˆe 1 .ˆe 2 = 0, eˆ 3 .eˆ 3 = 1,etc.
More generally, write this in the compact notation