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Electrostatic Potential for Two Concentric Cylindrical Shells

  1. Feb 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Two very long hollow conducting cylindrical shells are situated along the x-axis. The shells are concentric and have negligible thickness. The inner shell has a radius a and a linear charge density +lambda, while the outer shell has a radius b and a linear charge density -lambda. Take the zero of electrostatic potential to be at r = 0. The coordinate r measures the distance from the common axis of the two cylinders in a region far from either end.

    a) Determine the electrostatic potential V(r) for all values of r
    b) Sketch V(r) vs. r for all r
    c) Determine the potential difference DeltaV between r=a and r=b
    d) If a positive charge +q is released from rest at r=a, what will be its kinetic energy when it reaches the outer cylinder at r=b.


    2. Relevant equations
    V(r)=q/4piE0r
    Delta V=Vb-Va=q/4PiE0*(1/rb-1/ra)
    Ke=1/2mv^2
    U=kq1q2/r
    KEf=Ui


    3. The attempt at a solution
    a) at r=a
    V=q/4piE0a

    at r=b
    V=q/4piE0b

    b) as the radius increases the potential goes down. It starts at some positive y value and ends at some negative y value.

    c)Delta V=Vb-Va=q/4PiE0*(1/b-1/a)

    d) U=kq1q2/a
    1/2mv^2=kq1q2/r
    v^2=2kq1q2/mr
    v=(2kq1q2/mr)^1/2
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 14, 2010 #2
    I'm just wanting to know if i got this right
     
  4. Feb 14, 2010 #3

    ehild

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    The formula you quoted refers to a point charge. These are very long cylinders.

    ehild
     
  5. Feb 15, 2010 #4
    V(r)= 1/4piE0 Int dq/r
     
    Last edited: Feb 15, 2010
  6. Feb 15, 2010 #5
    Part C)

    EA=Qenc/E0
    E=lamda/PirE0
    Delta V=-lamda/PiE0 Int dr/r
    Delta V= lamda/PiE0 ln(b/a)
     
    Last edited: Feb 15, 2010
  7. Feb 15, 2010 #6

    ehild

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    You have started part C well, but it is still wrong.

    The first questions were:

    "a) Determine the electrostatic potential V(r) for all values of r
    b) Sketch V(r) vs. r for all r"

    So what is the potential for r<a? for r>b?

    ehild
     
  8. Feb 15, 2010 #7
    what equation am i supposed to use to figure out the potential? so confused.

    V (r) = 1/4PiE0 * Int dq/r ?
     
  9. Feb 15, 2010 #8

    ehild

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    Find the electric field first. What is it inside the inner cylinder? Between a and b? for r>b? Use Gauss' law.
    For r<0, the enclosed charge is 0. If a<r<b, the enclosed charge is +lambda *Length of the cylinder. For a cylinder with r>b, the enclosed charge is 0. E is the negative gradient of the potential. What is the potential like if E=0?

    ehild
     
    Last edited: Feb 15, 2010
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