# Electric field and total charge

uchicago2012

## Homework Statement

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.

Gauss' Law

## 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...

## Answers and Replies

cupid.callin
ß = A r tells density at any part at distance x from central axis.
this charge at x will be distributed along the circumference of that little part dx at distnce x

so dQ = dx * ß
where B will be A x

what is that p(r) ???

uchicago2012
p(r) is density as a function of r. Sorry I switched the symbols, I'm used to using rho. So is r a constant then? I thought it was a variable, because it said A was a constant and B was the density, so I thought B varied in accordance with r, which would be the distance from the central axis. Then the R would be the actual radius of the cylinder. But you think it's a constant?

cupid.callin
i didnt say its a constant ... did i?

i just used the expression to find charge dQ at distance x, now x is variable ...(thats why eqn include dx)

and dint worry about the radius stuff ... density given is linear

uchicago2012
Right, I see. So then Q = Ax2/2. But what does length L have to do with the integral? X is the distance from the central axis, so I don't see how to incorporate L into that.

cupid.callin
what will be the limits of integral?