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X
Now what do you suppose is going to happen when we have a volume
of charge? If we have a volume of charge, let's draw our X and
our Y. Z would come out this way. Excuse me. And let's have a
little bit of cube right here that has a charge in it. So the
charge right here is going to be a volume charged distribution
which is going to be 3 amps per meter cubed, and it's going to go
to the X equal 1, Y equal 1, Z equal 1 corners. The first thing
we do is we write RS. RS is the vector from the origin to a point
on my source, my current density. Oh, and I need to give a direction
to my current density-- let's let that be going in the Z direction.
So RS is equal to XS in the X direction, plus YS in the Y
direction, plus ZS in the Z direction. None of these are
constant. So I can't make any changes or simplifications at this
point. Let's find where we want to find the magnetic field.
Let's suppose that I want to find H right here out on the X axis
at the point where X is 4, Y is 0 and Z is 0. So RP is going to
be XP in the X direction, plus YP in the Y direction, plus ZP in
the Z direction. But XP is 4. YP is 0. And ZP is 0. So I can
simplify this one. Then RSP is RP minus RS, which is going to
give me 4 minus XS in the X direction, minus YS in the Y
direction, minus ZS in the Z direction. Let's define our current.
I in this case is going to be JV/DV. So that is going to be 3.
And then what's DV? DV is a small piece of this volume. So it's
going to be DX/DY and DZ. And since I know I'm going to be
integrating the source, those are the S terms. Now, I need a
direction here, and that is the direction of the current, Z. Now
let's do our I cross R term. And let's draw our 3-by-3 matrix.
Here's X. Here's minus Y. And here's Z. Plug in I on this row
and plug in R on this row. So I has only a Z term now. It is 3
DXS, DYS, DZS and 0 every place else. The R vector, that's right
here, is going to be 4 minus XS, minus YS and minus ZS. Now let's
do our cross-product. You take out the column and the row that X
is in and cross-multiply. Right here is 0, minus this value,
which is going to give me plus YS times 3 DXS, DYS, DZS. And
that's going to be in the X direction. The Y part is going to be,
again, cross-multiplying, it's going to be this minus that. 3
DXS, DYS, DZS, times 4 minus XS. Those minuses cancel out. Now
let's consider the Z term. The Z term, again, eliminate column
and row. Cross-multiply that 0 and this is 0. So the answer that
I get there is 0. Now, let's take I and take a good look at what
we've got here. In all cases we've got a vector X. Here's the
vector Y. We've got our volume components right there. Bring
those to the outside for our convenience. And now we're going to
apply Coulomb's Law. Sorry, not Coulomb's Law, Biot-Savart.