Capacitor Problem with distance-dependent dieletric

[rofldude188]

Homework Statement

Two plates of separation d and cross-section area A make a parallel capacitor. Capacitor is charged fully to charge Q. The battery is then disconnected and space between plates is filled with inhomogeneous dieletric material with dieletric permittivity of $$\epsilon = \epsilon_o(1 + \frac{y}{d})$$ where origin of y is on the bottom plate.

Relevant Equations

C = Q/V

a) Find Electric Field at any point in the dieletric in the terms of the parameters given
Making a pillbox Gaussian surface with one end in the conductor where E = 0 and the other end in the dieletric we have that $$\oint D \cdot dS = \rho_s A \implies D = \rho_s \implies E = \frac{Q}{A \epsilon_o(1 + \frac{y}{d})}$$

b) Calculate potential difference across dieletric
$$\Delta V = - \int_{0}^{d} E \cdot dL = \int_{0}^{d} Edy = \int_{0}^{d} \frac{Q dy}{A \epsilon_o(1 + \frac{y}{d})} = \frac{Q}{A \epsilon_o} d\ln(2) $$

c) Calculate capacitance of the capacitor
$$C = \frac{Q}{\Delta V} = \frac{A \epsilon_o}{d \ln2}$$

d) How much work is done to insert the dieletric

I have no idea how to do this, my guess would be it would be equal to $$W = \frac{Q^2}{2C}$$ because that is the energy stored in a capacitor? Can anybody verify what I have done is correct?

 

[kuruman]

Parts (a)-(c) look fine. For part (d) you can calculate the magnitude but not the sign of the "work done" because the problem does not specify what entity is doing the work, the electrical forces or the external agent who inserts the dielectric. One is the negative of the other. In any case the magnitude of the work is $$W=|U_{f}-U_{i}|=\left |\frac{Q^2}{2C_f}-\frac{Q^2}{2C_i}\right|$$where $C_f$ and $C_i$ are, respectively, the capacitances after and before insertion of the dielectric.

 

[rofldude188]

Parts (a)-(c) look fine. For part (d) you can calculate the magnitude but not the sign of the "work done" because the problem does not specify what entity is doing the work, the electrical forces or the external agent who inserts the dielectric. One is the negative of the other. In any case the magnitude is of the work is $$W=|U_{f}-U_{i}|=\left |\frac{Q^2}{2C_f}-\frac{Q^2}{2C_i}\right|$$where $C_f$ and $C_i$ are, respectively, the capacitances after and before insertion of the dielectric.

That makes complete sense thank you.