Intensity of polarization vector of a plate capacitor

[gruba]

Homework Statement

Plate capacitor with distance between plates 'd=4mm' is fully filled with homogeneous linear dielectric. Capacitor is connected to the source of DC voltage, first stationary state is established, then the capacitor is separated from DC voltage source. After that, dielectric is fully eliminated from capacitor and new stationary state is established. Difference between voltage in second and first stationary state is '(delta)U=144V'. Calculate intensity of polarization vector 'P' of a dielectric in the first stationary state (capacitor is separated from DC voltage source, and dielectric is not eliminated).

Homework Equations

((integral)DdS)=Q
D=e₀E+P
(delta)U=U₂-U₁

The Attempt at a Solution

Law of conservation of charge is valid, so Q=const.
In the first stationary state, we have a plate capacitor with dielectric, so we use Gauss law for dielectrics:
((integral)DdS)=Q
D=Q/S=e₀E+P (1)
In the second stationary state, we have a plate capacitor without dielectric (vacuum), so we use Gauss law for vacuum:
((integral)EdS)=Q/e₀
e₀E=Q/S (2)
Combining (1) and (2) gives
P=e₀(E₀-E)=e₀*(delta)U/d
I get the intensity of polarization vector P=318.6 nC/m²
What puzzles me is that we need to find intensity of polarization vector in the FIRST stationary state, so we need to find voltage U₁ in that state? Also, is there intensity of polarization vector if there are no dielectrics?

Thanks.

 

[Simon Bridge]

What puzzles me is that we need to find intensity of polarization vector in the FIRST stationary state, so we need to find voltage U1 in that state? Also, is there intensity of polarization vector if there are no dielectrics?

... there is no polarization without a dielectric to be polarized.
You do not need to know U1, as you discovered, because you know the difference in voltage between the states, but you can calculate it if you like.

 

[gruba]

... there is no polarization without a dielectric to be polarized.
You do not need to know U1, as you discovered, because you know the difference in voltage between the states, but you can calculate it if you like.

Could you please tell if intensity of polarization vector in the first stationary state is

P=-e₀*U₁/d

or

P=e₀*(delta)U/d

Which result is correct and why? Also, how to calculate U₁?
If P=P₁+P₂, P₂=0 because there is no dielectric, right?

 

[rude man]

.

I will rush in where angels fear to read and state that the problem is unsolvable.
EDIT: solvable!
Hint: D = ε₀E₁ + P = ε₀E₂
You know E₂ - E₁ from knowing V₂ - V₁.
Sorry about earlier post.

 

[Simon Bridge]

Relationship between electric displacement and the voltage for a parallel-plate capacitor?

 

[gruba]

Relationship between electric displacement and the voltage for a parallel-plate capacitor?

Could you tell if P=318.6 nC/m² is correct or not and why?

 

[rude man]

Could you tell if P=318.6 nC/m² is correct or not and why?

I can tell, and it's correct! Even the units. Congrats!

 

[Simon Bridge]

I don't normally comment on if an answer is correct or not.
Anyway - what if I say it is wrong? How do you know I have got it right and the rude man is not wrong again?

You are training to solve problems that nobody knows the answer to - if you get really good, you'll be solving problems nobody has even thought of yet - who will you ask then?

 

[rude man]

Bit severe, what?

Yes, once at work as a scientist or engineer you won't know the answers a priori. But by then you will hopefully have learned how to get them. As a student, by definition you still don't have that knowledge.

Which is why the typical textbook has numerical or other answers to all the odd or even problems, and why instructors are there to provide answers also. And why I make it a practice, time permitting, to have a numerical answer at hand should the OP want to compare with his/hers. If the answers don't agree I don't offer a guarantee mine is right but it usually is, thank you. Textbook answers have been known to be wrong also; we had an example here not long ago. Doesn't lessen their value.

Bottom line: don't expect perfection. We're all human.