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 1 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
Are $\rho_b$ and $\sigma_b$ due to real charges?
1. Of course not! They are as fictitious as it gets!
2. Of course they are! They are as real as it gets!
3. I have no idea
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