If you put a polarizable material (a dielectric) in an external field $\mathbf{E}_e$, it polarizes, adding a new field, $\mathbf{E}_p$
(from the bound charges). These superpose, making a total field, $\mathbf{E}_T$. What is the vector equation relating these three fields?
1. $\mathbf{E}_T + \mathbf{E}_e + \mathbf{E}_p = 0$
2. $\mathbf{E}_T = \mathbf{E}_e - \mathbf{E}_p$
3. $\mathbf{E}_T = \mathbf{E}_e + \mathbf{E}_p$
4. $\mathbf{E}_T = -\mathbf{E}_e + \mathbf{E}_p$
5. Something else
Note:
* CORRECT ANSWER: C
## Announcements
* Exam 2 (Wednesday, November 8th 7-9pm)
* Covers through Homework 9 (solutions posted after class)
* "Comprehensive" exam (need to remember old stuff)
* 1 sheet of your own notes; formula sheet posted
## What's on Exam 2?
* Using Legendre polynomials and separation of variables in spherical coordinates, solve for the potential and distribution of charge in a boundary value problem
* Using the multipole expansion, find the approximate form of the potential for a distribution of charge
* Determine the bound charge in a material with a given polarization
* Find the electric potential for a 1D Laplace problem and explain how you would determine it using the method of relaxation
* (BONUS) Solve a 3D Laplace problem in Cartesian coordinates
<img src="./images/doped_cylinder.png" align="right" style="width: 100px";/>
A solid non-conducting dielectric rod has been injected ("doped") with a fixed, known charge distribution $\rho(s)$. (The material responds, polarizing internally.)
When computing $D$ in the rod, do you treat this $\rho(s)$ as the "free charges" or "bound charges"?
1. "free charge"
2. "bound charge"
3. Neither of these - $\rho(s)$ is some combination of free and bound
4. Something else.
Note:
* CORRECT ANSWER: A
We define "Electric Displacement" or "D" field,
$$\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}$$
If you put a dielectric in an **external** field, it polarizes, adding a new **induced** field (from the bound charges). These superpose, making a **total** electric field. Which of these three E fields is the "E" in the formula for D above?
1. $\mathbf{E}_{ext}$
2. $\mathbf{E}_{induced}$
3. $\mathbf{E}_{tot}$
Note:
* CORRECT ANSWER: C
We define $\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}$, with
$$\oint \mathbf{D}\cdot d\mathbf{A} = Q_{free}$$
<img src="./images/charge_in_spherical_dielectric.png" align="right" style="width: 200px";/>
A point charge $+q$ is placed at the center of a dielectric sphere (radius $R$). There are no other free charges anywhere. What is $|\mathbf{D}(r)|$?
1. $q/(4 \pi r^2)$ everywhere
2. $q/(4 \varepsilon_0\pi r^2)$ everywhere
3. $q/(4 \pi r^2)$ for $r < R$, but $q/(4 \varepsilon_0\pi r^2)$ for $r>R$
4. None of the above, it's more complicated
5. We need more info to answer!
Note:
* CORRECT ANSWER: A
For linear dielectrics the relationship between the polarization, $\mathbf{P}$, and the total electric field, $\mathbf{E}$, is given by:
$$\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E}$$
where $X_e$ is typically a known constant. Think about what happens if (1) $X_e \rightarrow 0$ or if (2) $X_e \rightarrow \infty$. What do each of these limits describe?
1. (1) describes a metal and (2) describes vacuum
2. (1) describes vacuum and (2) describes a metal
3. Any material can gave either $X_e \rightarrow 0$ or $X_e \rightarrow \infty$
Note:
* CORRECT ANSWER: B