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Electric field and total charge

  • #1

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.

Homework Equations


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

  • #2
1,137
0
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) ???
 
  • #3
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?
 
  • #4
1,137
0
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
 
  • #5
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.
 
  • #6
1,137
0
what will be the limits of integral?
 

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