Paralle plate capacitor, LIH dielectric, fringing field

[Metaleer]

Hey, all.

If we partially introduce a linear, isotropic and homogeneous dielectric slab in a charged, isolated, parallel plate capacitor, we know that it experiences a forces pulling it into the dielectric, and we can obtain the expression of this force using energy considerations. However, when the energy calculation is done, we assume a uniform E field in the capacitor, always perpendicular to the dielectric, so in theory it looks like the E field can't pull on any charge that's in the dielectric to pull it in, yet the energy calculation still reveals a force. Books then say it's the fringing E field outside the capacitor that's pushing the dielectric in, but this fringing field wasn't taken into account when getting the energy!

What's going on? How does this method self correct itself?

Thanks in advance.

 

[ZapperZ]

That is a very good question. In fact, this was carefully addressed in an AJP paper many years ago:

"Force on a Dielectric Slab Inserted into a Parallel-Plate Capacitor", S. Margulies (Am. J. Phys. v.52, p.515 (1984)).

In it, he wrote this:

For example, how can the force act to pull the slab into the volume between the plates when the electric field there is perpendicular to this direction? If this is explained - the force is, of course, due to the fringe field - an apparent paradox arises: How can the virtual-work calculation yield an answer when it is explicitly based on the assumption of a uniform electric field existing only in the region between the plates, and so does not include the fringe field at all?

Sounds familiar? :)

Zz.

 

[Metaleer]

That is a very good question. In fact, this was carefully addressed in an AJP paper many years ago:

"Force on a Dielectric Slab Inserted into a Parallel-Plate Capacitor", S. Margulies (Am. J. Phys. v.52, p.515 (1984)).

In it, he wrote this:



Sounds familiar? :)

Zz.

Woa, that is exactly what I needed! It's a shame I can't access that article, though.

But it's kind of incredible that what I asked turned out to be something that only a research article could answer.

Thanks for the help, ZapperZ.

 

[Meir Achuz]

For some reason, my download from AJP is not working, but there is a simple answer to your question.
In the energy calculation, the end of the dielectric slab moves an infinitesimal distance, but is well within the plates where the field is uniform. The calculated force then does not depend on what happens at the edge of the plates. The energy is given by an integral of E.D over all space, so the fringing field would affect the total energy, but it does not affect the change in energy caused by the infinitesimal displacement
If the energy calculation were attempted for the edge of the dielectric in the fringing field, the calculation would be more difficult, and the force would be different.

 

[sardar]

Hello,

Thank you for your question. It is true that when calculating the force on a dielectric slab placed in a parallel plate capacitor, we assume a uniform electric field inside the capacitor. However, this assumption is only valid in an ideal scenario where the plates are infinite in size and the dielectric slab is very small compared to the distance between the plates. In reality, the plates have finite size and the dielectric slab may not be small enough to be considered as a point charge.

In such cases, the electric field lines do not remain perfectly perpendicular to the dielectric slab and there is a fringing field present at the edges of the plates. This fringing field exerts a force on the dielectric slab, pushing it into the capacitor. This force is accounted for in the energy calculation, even though it may not be explicitly mentioned.

In other words, the energy calculation takes into account the fringing field and the resulting force on the dielectric slab, even though it may not be explicitly stated. This is why the method self-corrects itself and gives an accurate result for the force on the dielectric slab.

I hope this helps to clarify the confusion. Let me know if you have any further questions.