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Bound surface charge - hollow tube

  1. Sep 3, 2014 #1
    1. The problem statement, all variables and given/known data
    A thick-walled cylindrical tube of dielectric material has internal radius a/2 and external radius 2a, and its polarization is given in cylindrical coordinates by the expression P = (A/r2) er.

    Derive expressions for the net charge on the inner and outer surfaces of a length L of the tube and for the charge within the volume of length L of the tube.


    2. Relevant equations
    σb = P . n hat
    where n hat is the outward-pointing unit vector normal to the surface and P is the polarization.
    surface area of cylinder (ignoring ends) = 2∏rL


    3. The attempt at a solution

    σb total = σb inner + σb outer

    σb outer = (A/(2a)2) * area of cylinder
    σb outer = (A/(2a)2) * 2∏(2a)L
    σb outer = ∏AL/a

    Now the bit I can't seem to marry with the model answer . . .

    σb inner = (A/(a/2)2) * 2∏(a)L
    σb inner = (4A/a2) * 2∏(a)L
    σb inner = 8∏AL/a

    At least that's how I got it.

    The model answer shows it to be
    σb inner = 4∏AL/a

    It has to be a problem I'm having with algebra again, but I can't see where.
    Could you show me where I went wrong please.

    Or am I right?
     

    Attached Files:

    Last edited: Sep 3, 2014
  2. jcsd
  3. Sep 3, 2014 #2
    Aaaaah!
    Details.
    Please ignore this.

    Yes, the wrong bit is me not looking proplerly.
    radius is a/2 not a.
    Sorry.
     
  4. Sep 3, 2014 #3

    BvU

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    Still has me puzzled. In the attachment I read ##\vec P = \left ( A/r^3\right ) \, \vec e_r ## (in the text; in the picture it's hard to distinguish if it's a 3 or a 2).

    Or is your exercise another one, with the a/2 instead of the a and the 2 instead of the 3 ?
     
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