Lucite problem
Lucite, or 'plexiglass' is a lightweight, transparent plastic, that during Covid was being used as a "sneeze guard" in grocery stores, election polling stations, and hospitals to protect people who work with the public.
...bombarded by electrons
When a beam of high-energy electrons bombards an insulating material such as lucite, the electrons penetrate into the material, come to rest at a depth which depends on their initial kinetic energy, and remain trapped inside. This is "free charge" in Griffiths' sense: It has been placed in the system by some "mysterious outside agent" - in this case, the bombarding electron beam. As a result of its presence the material may become polarized, resulting in various "bound" charge densities.
Consider an initially unpolarized, electrically neutral block of lucite (with dielectric constant
$K=3.2$), of dimensions 5 cm $\times$ 5 cm $\times$ 1.2 cm. See the sketch below, but note that the 1.2 cm dimension--let's call this the $\hat z$ direction--is stretched out, and not to scale with the the other dimensions.
Now, imagine that this block of lucite has been bombarded with electrons incident from above the block.
- The bombarding beam is an electric current of 0.1 microamperes (1 amp = 1 Coulomb / sec).
- The beam was turned on for 1 second, and then turned off.
- The beam was spread uniformly across the top surface of the block.
- All the electrons in the beam were trapped
6 mm below the surface in a region 2 mm thick:
Let's say that the 2 mm layer with electrons is centered at the origin, at $z=0$. The block is positioned in the region $-6$ mm $\lt z \lt +6$; $-2.5$ cm $\lt x \lt +2.5$; $-2.5$ cm $\lt y \lt +2.5$;
- After bombardment, conducting plates are attached to the top and bottom surfaces of the block, and are grounded (V=0).
Assume a uniform density for the trapped electrons in the 2 mm thick layer, and neglect edge effects--that is, we're going to approximate all the fields as if this were an infinite slab of lucite in the $x$ and $y$ directions. So, you may assume that all the fields are independent of $x$ and $y$, and only depend on $z$. This should be a pretty good approximation as long as we consider $z$ distances that are small compared to the width of the system. That is $$z\ll 5\text{ cm}=50\text{ mm}.$$
- What is the free-charge density in the region containing the electrons? (In units of Coulombs / m^3).
- Using Gauss' law for the displacement field, and the symmetry of this system, choose some useful Gaussian surface (or "Gaussian pillbox" and find $\myv D(z)$ everywhere inside the lucite block. That is, for $-6$ mm$\lt z \lt$ +6mm. Describe and/or sketch the Gaussian surface you used. (A side view, say a cross sectional view of your Gaussian pillbox in $x$ and $z$ might be useful, so I can see the top and bottom of your pillbox in the context of the lucite block.)
-
Free charge gives rise to a bound charge density.
Contemplate this passage in Griffiths at the beginning of section 4.4.2 (p 186):
In a homogeneous linear dielectric the bound charge density ($\rho_b$) is proportional to the free charge density ($\rho_f$) $$\rho_b=-\myv\grad\cdot \myv P=-\myv \grad\cdot\left( \epsilon_0\frac{\chi_e}{\epsilon}\right)\myv D=-\left(\frac{\chi_e}{1+\chi_e} \right)\rho_f.$$
Treating lucite as a homogeneous linear dielectric: Qualitatively, what are the bound charge densities (negative, positive, zero) in these regions...
- $|z|\lt $ 1 mm.
- 1 mm $\lt |z| \lt$ 6 mm.
- Now, quantitatively, using the dielectric constant of lucite... Find the bound charge density, $\rho_b(z)$, in all regions of the lucite block. (Check the sign of your bound-charge density using your physical intuition about unlike charges attracting...)
- Find the bound surface charge density, $\sigma_b$, on the top surface and bottom surfaces by reasoning that... if the lucite started out electrically neutral, the total charge on these 2 surfaces must be equal and opposite the integrated bound charge.
- Find $\myv E(z)$ and $\myv P(z)$. (And make qualitative sketches of $D_z(z)$, $E_z(z)$, and $P_z(z)$.
- What is the magnitude of the electric field outside of the region containing the electrons? (In units of V/m)
- Find the electric potential, $V(z)$. Check that the magnitude of the voltage at $z=0$ is close to 4 kV. Sketch $V(z)$.
- Qualitatively, how would the graph of $V$ be affected if the sheet of electrons were closer to one face of the block than to the other? Make sketch.
- What is the total energy stored in the block? What is the danger if the block explodes? (For example, compare the energy density with this chart where 'MJ/L' means 'Megajoules per liter'.)