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.

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...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 $$z\ll 5\text{ cm}=50\text{ mm}.$$

  1. What is the free-charge density in the region containing the electrons? (In units of Coulombs / m^3).
  2. 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.)
  3. 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.
  4. 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...)
  5. 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.
  6. Find $\myv E(z)$ and $\myv P(z)$. (And make qualitative sketches of $D_z(z)$, $E_z(z)$, and $P_z(z)$.
  7. What is the magnitude of the electric field outside of the region containing the electrons? (In units of V/m)
  8. Find the electric potential, $V(z)$. Check that the magnitude of the voltage at $z=0$ is close to 4 kV. Sketch $V(z)$.
  9. 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.
  10. 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'.)