Two different dielectrics between parallel-plate capacitor

In summary, the presence of two different dielectrics between the plates of a parallel-plate capacitor affects its capacitance by altering the electric field distribution and the overall dielectric constant. Each dielectric material contributes differently based on its permittivity, leading to a combined effect that can be calculated using the geometry of the capacitor and the properties of the materials. This configuration can be analyzed using series and parallel models, impacting the capacitor's energy storage capacity and performance in electrical circuits.
  • #1
zenterix
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Homework Statement
Two dielectrics with dielectric constants ##\kappa_1## and ##\kappa_2## each fill half the space between the plates of a parallel-plate capacitor as shown in the figure below.

Each plate has an area ##A## and the plates are separated by a distance ##d##.

Compute the capacitance of the system.
Relevant Equations
##\oint_S\vec{E}\cdot\hat{n} dS=\frac{Q_{\text{enc}}}{\epsilon_0}##
We have a parallel plate capacitor with two different dielectrics

1706202697499.png


It seems to be the case that the potential difference on each half of the capacitor is the same.

Initially, the electric field was ##\vec{E_0}=\frac{2\sigma_+}{\epsilon_0}\hat{j}##.

If we were to insert a single dielectric material with dielectric constant ##\kappa_e## between the plates, this electric field would weaken to ##\frac{E_0}{\kappa_e}##.

The potential difference would also decrease to ##\frac{|\Delta V_0|}{\kappa_e}##.

But now we have two halves.

If the potential difference is the same in the two halves, then it must be that the electric fields are the same in the two halves. But then the charges on the halves must differ.

Does the reason the potential difference is the same on the two halves arise because of the path independence of ##\vec{E}##?
 
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  • #2
zenterix said:
Does the reason the potential difference is the same on the two halves arise because of the path independence of ##\vec{E}##?
The potential difference is the same for the two halves because of the electrostatic properties of conductors. For any conductor in electrostatic equilibrium, what can you say about the potential at two different points of the conductor?

Suppose you have two conductors that are each in electrostatic equilibrium. What can you say about the potential difference between any point of one conductor and any point of the other conductor? Does it depend on the choice of points?
 
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  • #3
TSny said:
The potential difference is the same for the two halves because of the electrostatic properties of conductors. For any conductor in electrostatic equilibrium, what can you say about the potential at two different points of the conductor?

Suppose you have two conductors that are each in electrostatic equilibrium. What can you say about the potential difference between any point of one conductor and any point of the other conductor? Does it depend on the choice of points?
The potential inside each conductor is constant. Thus, the potential difference between any point inside one conducting plate and any point inside the other conducting plate is constant.

Suppose that instead of conductors we had charged sheets as capacitor plates.

Now what is it that prevents the potential difference from being different in each half of the capacitor?
 
  • #4
If the potential difference is the same......as we've established. Are the two different capacitors in series or parallel?

Also conductors by definition are equipotentials.
 
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  • #5
zenterix said:
The potential inside each conductor is constant. Thus, the potential difference between any point inside one conducting plate and any point inside the other conducting plate is constant.
Yes.

zenterix said:
Suppose that instead of conductors we had charged sheets as capacitor plates.

Now what is it that prevents the potential difference from being different in each half of the capacitor?
If the charge density is fixed on each sheet so that the charge cannot move around on either sheet, then the electric field inside the two dielectrics would be different. So, the potential difference between the sheets would be different for the two halves. We would not call this system a capacitor, since the sheets are not conductors.
 
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  • #6
PhDeezNutz said:
Also conductors by definition are equipotentials.
Not exactly by definition but when conductors are in steady state or electrostatic equilibrium.

In a transient state that is when there is current density inside the conductor that is different from zero then they might not be equipotential even if they are perfect conductors. The case of a perfect conductor that is shaped as a coil and has time varying current density should come up here but I am opening a can of worms now. For anyone who is interested can take a look at this thread https://www.physicsforums.com/threads/inducing-emf-through-a-coil-understanding-flux.940861/.
 
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  • #7
Delta2 said:
Not exactly by definition but when conductors are in steady state or electrostatic equilibrium.

In a transient state that is when there is current density inside the conductor that is different from zero then they might not be equipotential even if they are perfect conductors. The case of a perfect conductor that is shaped as a coil and has time varying current density should come up here but I am opening a can of worms now. For anyone who is interested can take a look at this thread https://www.physicsforums.com/threads/inducing-emf-through-a-coil-understanding-flux.940861/.

I’m still a n00b dealing with electrostatics but your post has given me things to think about as I advanced my studies.

Thanks!
 
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FAQ: Two different dielectrics between parallel-plate capacitor

What is a parallel-plate capacitor with two different dielectrics?

A parallel-plate capacitor with two different dielectrics is a capacitor where the space between its plates is filled with two distinct dielectric materials. These materials have different dielectric constants, and they can be arranged either in series or parallel within the capacitor.

How do you calculate the capacitance of a parallel-plate capacitor with two different dielectrics in series?

For dielectrics in series, the total capacitance can be calculated using the formula: \[ \frac{1}{C} = \frac{d_1}{\epsilon_1 A} + \frac{d_2}{\epsilon_2 A} \]where \( d_1 \) and \( d_2 \) are the thicknesses of the two dielectric layers, \( \epsilon_1 \) and \( \epsilon_2 \) are their respective dielectric constants, and \( A \) is the area of the plates.

How do you calculate the capacitance of a parallel-plate capacitor with two different dielectrics in parallel?

For dielectrics in parallel, the total capacitance can be calculated using the formula: \[ C = \frac{\epsilon_1 A_1}{d} + \frac{\epsilon_2 A_2}{d} \]where \( A_1 \) and \( A_2 \) are the areas of the plates covered by the two dielectrics, \( \epsilon_1 \) and \( \epsilon_2 \) are their respective dielectric constants, and \( d \) is the separation between the plates.

What effect do the different dielectric materials have on the overall performance of the capacitor?

The presence of different dielectric materials affects the overall capacitance, energy storage capacity, and breakdown voltage of the capacitor. The effective dielectric constant is a weighted average based on the arrangement and properties of the dielectrics, which influences the capacitor's performance characteristics.

Why might one use two different dielectrics in a parallel-plate capacitor?

Using two different dielectrics can optimize the capacitor for specific applications. For instance, one dielectric might provide a higher dielectric constant to increase capacitance, while the other offers better thermal stability or higher breakdown voltage. This combination can enhance the overall performance and reliability of the capacitor in varying conditions.

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