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rofldude188

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- 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?

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?