Consider a cylinder of radius $a$ and height $b$ that has it base at the origin and is aligned along the $z$-axis. The polarization of this cylinder is "baked in" and can be modeled using $$\mathbf{P} = P_0 \left(\dfrac{z}{b}\right)\hat{z}.$$ Determine the total dipole moment of this cylinder: 1. $P_0 \pi a^2 b \hat{z}$ 2. $\frac{1}{2} P_0 \pi a^2 b \hat{z}$ 3. $P_0 2 \pi a b^2 \hat{z}$ 4. $\frac{1}{2}P_0 \pi a b^2 \hat{z}$ 5. Something else Note: * Correct answer: B take the integral
## Exam 2 Information * Covers through polarization (up to Ch 4.2.3) * Emphasizes material since Exam 1 * But don't forget Exam 1 material! * Specifics on Wednesday
In the following case, is the bound surface and volume charge zero or nonzero? <img src="./images/mini_dipoles_matter_1.png" align="center" style="width: 400px";/> 1. $\sigma_b = 0, \rho_b \neq 0$ 2. $\sigma_b \neq 0, \rho_b \neq 0$ 3. $\sigma_b = 0, \rho_b=0$ 4. $\sigma_b \neq 0, \rho_b=0$ Note: * CORRECT ANSWER: D
In the following case, is the bound surface and volume charge zero or nonzero? <img src="./images/mini_dipoles_matter_2.png" align="center" style="width: 400px";/> 1. $\sigma_b = 0, \rho_b \neq 0$ 2. $\sigma_b \neq 0, \rho_b \neq 0$ 3. $\sigma_b = 0, \rho_b=0$ 4. $\sigma_b \neq 0, \rho_b=0$ Note: * CORRECT ANSWER: B
A VERY thin slab of thickness $d$ and area $A$ has volume charge density $\rho = Q / V$. Because it's so thin, we may think of it as a surface charge density $\sigma = Q / A$. <img src="./images/thin_slab_polarization.png" align="center" style="width: 400px";/> The relation between $\rho$ and $\sigma$ is: 1. $\sigma = \rho$ 2. $\sigma = \rho d$ 3. $\sigma = \rho/d$ 4. $\sigma = V \rho$ 5. $\sigma = \rho/V$ Note: * CORRECT ANSWER: B
A dielectric slab (top area $A$, height $h$) has been polarized, with $\mathbf{P}=P_0$ in the $+z$ direction. What is the surface charge density, $\sigma_b$, on the bottom surface? <img src="./images/slab_p0_polarization.png" align="right" style="width: 400px";/> 1. 0 2. $-P_0$ 3. $P_0$ 4. $P_0 A h$ 5. $P_0 A$ Note: * CORRECT ANSWER: B
A dielectric sphere is uniformly polarized, $$\mathbf{P} = +P_0\hat{z}$$ What is the surface charge density? <img src="./images/sphere_p0_dielectric.png" align="right" style="width: 300px";/> 1. 0 2. Non-zero Constant 3. constant*$\sin \theta$ 4. constant*$\cos \theta$ 5. ?? Note: * CORRECT ANSWER: D
A dielectric sphere is uniformly polarized, $$\mathbf{P} = +P_0\hat{z}$$ What is the volume charge density? <img src="./images/sphere_p0_dielectric.png" align="right" style="width: 300px";/> 1. 0 2. Non-zero Constant 3. Depends on $r$, but not $\theta$ 4. Depends on $\theta$, but not $r$ 5. ? Note: * CORRECT ANSWER: A