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