Charge is distributed through an infinitely long cylinder of radius R in such a way that the charge density is proportional to the distance from the central axis: ß = A r, where A is a constant and ß is the density.
(a) Calculate the total charge contained in a segment of the cylinder of length L.
(b) Calculate the electric field for points outside the cylinder.
(c) Calculate the electric field for points inside the cylinder.
The Attempt at a Solution
so for a)
dQ = p(r) V dr
right? or is it
dQ = p(r) A dr
I can't decide. It says charge is distributed throughout the cylinder but in my book we always seem to use surface areas so I'm confused.
But if I go with my first equation and integrate to find the total charge, Q
Q = integral (A * r * pi * R2 * L * dr)
where r = distance of point from central axis and R = radius of cylinder
then = A * pi * R2 * L * integral r dr
which = (A * pi * R4 * L)/2
I integrated from 0 to R which leaves out half of the cylinder so I'm not so sure what to do about that... should I integrate to 2R intsead?
As far as for b and c, I can't see why they'd be different. My book talks about cylindrical symmetry and says
E = linear charge density/(2 * pi *e0 * r)
which is the electric field E due to an infinitely long, straight line of charge, at a point that is a radial distance r from the line. So wouldn't they be the same, with different r's? Perhaps for the one inside the surface, the density B = A * r would cancel with the r on the bottom. Or instead of this equation, should I go back and say:
e0 * E * A = q
and use my q from before, if it's even right?
And should I use the area or the volume then, for this part?
Ugh this electric stuff confuses me. I miss gravity...