A parallel-plate capacitor has an initial charge q and a plate separation distan

Question

A parallel-plate capacitor has an initial charge q and a plate separation distance d. (For both questions below, please make sure that your explanation arguments in terms of words and/or the mathematical symbols are clear and understandable) (a) If the capacitor is isolated, how much work must you do, in terms of q, d and plate area A, to increase the separation distance to ? Here, S > 1 is a positive, dimensionless quantity. Express your answer in terms of, the permittivity constant some or all of the variables q, d, and A. (b) If the capacitor is connected to a battery throughout, what's the change in the energy stored in the capacitor when the plate separation is increased from d to ? Here, > 1 is a positive, dimensionless quantity. Express your answer in terms of B, the permittivity constant, some or all of the variables q, d, and A. The answer should also clearly whether the change is positive. (i.e. the energy is increased) or negative.

Explanation / Answer

(A) C = e0 A / d

initial energy , U = q^2 / 2 C

now d -> Bd

then C' = e0 A / (B d) = C / B

final energy , U' = q^2 / (2 C / B) =B( q^2 / 2 C)

U' = B U

Work done = U' - U = (B - 1) U

= ( B - 1) q^2 / (2 e0 A / d)

( B - 1) d q^2 / (2 e0 A)

(B) now V = constant

Ui = C V^2 /2 = e0 A v^2 / 2 d

Uf = (e0 A / B d) V^2 / 2

Work done = Uf - Ui

= (e0 A V^2 / 2 d) [1/B - 1]

change will be negative,

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