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Capacitors and Parallel Conducting Plates

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The question asks me to consider a capacitor consisting of two parallel conducting plates 1 and 2 with a dielectric material between them. (full description of system in attached file) There is also a free charge density between the plates. the question then asks me to show that the potential in the region a<z<2a (a point between the plates) is shown to be a given equation (shown in file) so basically a system is described and an potential equation is supplied and I have to show how that equation is derived? I have supplied all the relevant associated equations I can think of that may be required I just don't know how to bring them all together to produce the required (given) equation. Please can you show me your working. Thank you.


Consider a capacitor consisting of two parallel conducting plates, 1 and 2, separation a, with a dielectric material of constant relative permittivity ε between the plates. Plate 1 is in the z = a plane and is held at a fixed potential V1, whereas plate 2 is in the z = 2a plane and is held at the fixed potential V2. The plates are so large compared to their separation that the electric field between them can be considered to depend only on z. Between the plates there is a free charge density where K is a constant.

Show that the potential in the region a < z < 2a is,

Hence determine the electric field E in the region a < z < 2a between the plates.

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The expert examines parallel plate questions.