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I Inserting a conductor in a parallel-plate capacitor

  1. Nov 5, 2017 #1
    Consider a parallel plate capacitor formed by two plates of length ##L## and width ##d##, separated by a distance ##e##. There is a vacuum in between the plates. Let's note the capacitance of this arrangement ##C_0##.

    I insert a conducting plate of length ##l=L/2##, with ##D##, and thickness ##e' <<e##. The position of the plate is measured by its ##(x,y)## coordinates, as shown below:
    uKSaN.jpg

    I would like to find the equivalent capacitance of this apparatus in terms of the distance ##x##.

    Of course if ##x<0##, the conductor is not inserted at all so the capacitance remains unchanged, ##C_0##.

    Consider the case where the conductor is inserted partially, i.e ##0<x<l##.

    According to my notes, in this case the apparatus is equivalent to the arrangement of capacitors below:

    34tCi.jpg

    where

    ##C_1=\frac{\epsilon_0Dx}{e-y-e'}##

    ##C_2=\frac{\epsilon_0Dx}{y}##

    ##C_3=\frac{\epsilon_0D(L-x)}{e}##

    **I do not understand why this configuration is equivalent to the arrangement of capacitors given above.**

    I guess ##C_1## is the capacitor formed by the top plate and the conductor, ##C_2## the capacitor formed by the bottom plate and the conductor, and ##C_3## the capacitor formed by the conductor itself. However this leaves me confused as the capacitance for the conductor should then be:

    ##C_3=\frac{\epsilon_0Dx}{e}##

    Finally, if we now consider the case where the conductor is fully inserted, i.e ##l<x<L##, then apparently the capacitor arrangement changes completely and we now actually have four capacitors (2 in series, which are parallel with the other two). I don't understand why.
     
  2. jcsd
  3. Nov 5, 2017 #2

    BvU

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    Hi,

    As Sherlock Holmes would say: then the logical conclusion is that your guess was wrong ....

    Look at ##C_3##: is your guess logical if e.g. ##x=0## ?
     
  4. Nov 6, 2017 #3
    As BvU has suggested, the issue is with your interpretation of ##C_{3}##. The capacitor ##C_{3}## is considered to be the capacitance of the space to the right of the conductor, leading to the form of ##C_{3}## given in your notes.
     
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