Two different dielectrics between parallel-plate capacitor

[zenterix]

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

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 $\vec{E}$?

 

[TSny]

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?

 

[zenterix]

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?

 

[PhDeezNutz]

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.

 

[TSny]

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.

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.

 

[Delta2]

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/.

 

[PhDeezNutz]

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!