Energy to make a dielectric explode?

[scottKC]

i'm working through a homework problem. it has us jump through a few hoops. the premise is that a bunch of electrons are injected into a block of lucite (relative permittivity is supplied). the electrons are all concentrated in a given volume within the block. it asks us for the volume bound charge density in the region of electrons (i'm assuming it is 0, since the electron density is uniform). then asks us for the bound surface charge density on the block (i calculated the free surface charge density at the interface between the region of electrons, then the bound surface charge density in the electron-free region from there).

anyways, the last part of the question is: "What is the energy stored in the block? Could the block explode?"

I'm lost at how to determine this. does anyone have any hints on the physics involved in determining when a dielectric will explode?

thanks in advance.

 

[quasar987]

What does the uniformity of the electron density has to do with there being no bound charge?

When there is free charge in a dielectric, the dipoles of the dielectric will rearange themselves around the charge, which results in a net bound charge density. Instead of guessing it was 0, why didn't you calculate it? you have all you need:

$$-\nabla \cdot \vec{P}=\rho_b$$
$$\vec{P}=\epsilon_0\chi_e \vec{E}=(\epsilon -\epsilon_0)\vec{E}$$
$$\nabla \cdot \vec{E}= \rho_f/\epsilon$$

 

[scottKC]

I'm guess I'm just having trouble with this. you're clearly right about the free charge density inducing a bound charge density within the volume injected with electrons (easy to find knowing the free charge density and the relative permittivity). i can find E outside of the electron volume knowing the free charge density. Then with E i can find P (again with the relative permittivity). With P i can find the bound charge density within the electron-free region. I can also find the bound surface density at the face of the block (P dotted with the normal vector).

am i thinking along the correct lines here?

 

[quasar987]

I'm guess I'm just having trouble with this. you're clearly right about the free charge density inducing a bound charge density within the volume injected with electrons (easy to find knowing the free charge density and the relative permittivity). i can find E outside of the electron volume knowing the free charge density.

How are you planing on doing this? Or is there a special geometry to this whole thing that allows for an easy determination of the field knowing the charge distribution?

 

[scottKC]

i have the dimensions of the block. it has a 25 square cm face, is 12 mm thick and the electrons are evenly distributed in a 2 mm thickness 6 mm below the surface. now that i think about it, maybe it won't be easy to determine E. i guess I'm missing the connection between a known volume charge density and the surface charge density that it induces. all the examples I've seen involve a conductor and a dielectric together, which is much easier since the conductor itself has a surface charge density.

i get the feeling I'm making this way too difficult on myself. but I'm having a rough time getting a good feeling for dielectrics. maybe if someone could explain what physically will happen in this situation i'll be able to figure it out on my own.

 

[quasar987]

Since the block is much wider than it is tall, you can make the approximation that the slab of charges in the middle produces a field oriented entirely in the z direction. (i.e. you can neglect the border effects and say that at the surface of the block, the field of the slab is pretty much the same as the field of an infnite plane)

In that case, it will be easy to find the field at the surface of the block (and hence P, and hence $\sigma_b$ by use of gauss's law.