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Now let's consider line integral and cylindrical coordinates.
Let's do it the same way, the same process that we did when we did
the rectangular coordinate system. In fact, let's consider the
exact same piece of wire that has four coulombs per meter charge
on it. And I want to know its total charge over length from 0 to
4 meters. The way I did that before was I was going to do a
integral. Line integral is only 1. And I need to put my element
over here DL. Is this going to be a scalar value or a vector
value? Since this is scalar, then my DL is also going to be a
scalar value. Now I need to decide which part of DL to use. This
is my DL component right here. And to do that, I'm going to
decide what is changing. And let's do that in spherical
coordinates. Over the length of this line right here is the R
value changing? Here's the value R, R, R, R. The radius of this
wire from the center is always the same. So DR is equal to 0.
How about D phi? Here is phi and this wire is always here at phi
equal 90 degrees. So D phi is also equal to 0. Does Z change?
Yes. Over the length of this line, DZ is not equal to 0. Now
let's go look at our line terms. Is our DR equal to 0? Yes. Our
DR is 0. And right here our D phi term is 0. So this value right
here is the only one that I'm going to be using for my DL and I
need the scalar part. So DL is equal to DZ. I'm going to put my
function four coulombs per meter here and I'm going to integrate
from Z equals 0 to 4 meters. So I'm going to get 4 coulombs per
meter times Z from 0 to 4, which is going to give me 16 coulombs
per meter just as it did before. Now let's imagine what would
have happened if I had been integrating a wire that was shaped
like an arc that was a quarter of a moon like this. So let's do
that one actually in red. Just as an example. So this arc right
here, instead of integrating over DZ, what would be changing? D
phi would be the term that was changing. So my integral right
here would have an RD phi component and let's say that this is
also four coulombs per meter in here and now let's see, phi is
going to vary from 0 right here is phi equals 0 and here is phi
equals 90 degrees. So from phi equal to 0 to phi equals 90
degrees and I need to multiply that by R. So instead of --
actually, instead of 90 degrees, let me use pi by 4 and now I need
an R value. Let's say that R is 1 meter. Let's plug that in
there. So I'm going to end up with 4 coulombs per meter times R,
which is 1 meter, times phi, from 0 to pi by 4. So when I get my
answer it's going to be 4 times pi by 4 coulombs, or pi coulombs
for this little piece of line.