Capacitor Problem with distance-dependent dieletric

In summary, the electric field at any point in the dielectric can be calculated using the given parameters by creating a pillbox Gaussian surface with one end in the conductor and the other in the dielectric. The potential difference across the dielectric can be calculated by integrating the electric field over the distance of the dielectric. The capacitance of the capacitor can be found by dividing the charge by the potential difference. The work done to insert the dielectric can be calculated as the difference in energy stored in the capacitor before and after insertion. The magnitude of the work is given by |U_f - U_i|, where U_f and U_i are the energies after and before insertion, respectively.
  • #1
rofldude188
5
0
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?
 
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  • #2
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.
 
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  • #3
kuruman said:
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.
 

1. What is a distance-dependent dielectric?

A distance-dependent dielectric is a material that exhibits variations in its dielectric properties based on the distance between the two conductive plates of a capacitor. This distance-dependent behavior can affect the capacitance of the capacitor and can be caused by factors such as the polarization of the dielectric material or the presence of air gaps.

2. How does distance affect the capacitance in a capacitor with a distance-dependent dielectric?

The capacitance in a capacitor with a distance-dependent dielectric is directly proportional to the distance between the two conductive plates. As the distance increases, the capacitance decreases due to a decrease in the electric field strength between the plates.

3. What are some examples of distance-dependent dielectric materials?

Some examples of distance-dependent dielectric materials include air, vacuum, and certain types of polymers and liquids. These materials exhibit changes in their dielectric properties as the distance between the plates of a capacitor changes.

4. How does the dielectric constant of a distance-dependent dielectric affect the capacitance?

The dielectric constant of a distance-dependent dielectric can greatly impact the capacitance of a capacitor. A higher dielectric constant results in a higher capacitance, as it indicates a stronger ability of the material to store electric charge. Conversely, a lower dielectric constant leads to a lower capacitance.

5. How can the distance-dependent dielectric effect be mitigated in capacitor design?

The distance-dependent dielectric effect can be mitigated in capacitor design by using materials with a lower dielectric constant, such as ceramic or mica, which exhibit more stable dielectric properties regardless of the distance between the plates. Another approach is to minimize air gaps and ensure a uniform distance between the plates to reduce the impact of distance on capacitance.

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