Which of the following are vectors? (I) Electric field, (II) Electric flux, and/or (III) Electric charge 1. I only 2. I and II only 3. I and III only 4. II and III only 5. I, II, and II Note: * CORRECT ANSWER: A

## Gauss' Law <img src="./images/gauss_pt_charge.png" align="center" style="width: 350px";/> $$\oint_S \mathbf{E}\cdot d\mathbf{A} = \int_V \dfrac{\rho}{\varepsilon_0}d\tau$$

<img src="./images/cubical_box.png" align="right" style="width: 350px";/> The space in and around a cubical box (edge length $L$) is filled with a constant uniform electric field, $\mathbf{E} = E_0 \hat{y}$. What is the TOTAL electric flux $\oint_S \mathbf{E} \cdot d\mathbf{A}$ through this closed surface? 1. 0 2. $E_0L^2$ 3. $2E_0L^2$ 4. $6E_0L^2$ 5. We don't know $\rho(r)$, so can't answer. Note: * CORRECT ANSWER: A * All the incoming flux on the left side comes out the right side

A positive point charge $+q$ is placed outside a closed cylindrical surface as shown. The closed surface consists of the flat end caps (labeled A and B) and the curved side surface (C). What is the sign of the electric flux through surface C? <img src="./images/ABC_cylinder.png" align="center" style="width: 600px";/> 1. positive 2. negative 3. zero 4. not enough information given to decide Note: * CORRECT ANSWER: B * This is meant to be hard to visualize, next slide illustrates it better.

Let's get a better look at the side view. <img src="./images/ABC_cylinder_side.png" align="center" style="width: 350px";/>

A positive point charge $+q$ is placed outside a closed cylindrical surface as shown. The closed surface consists of the flat end caps (labeled A and B) and the curved side surface (C). What is the sign of the electric flux through surface C? <img src="./images/ABC_cylinder.png" align="center" style="width: 600px";/> 1. positive 2. negative 3. zero 4. not enough information given to decide Note: * CORRECT ANSWER: B * Some of the incoming flux through C goes out A and B.

Which of the following two fields has zero divergence? | I | II | |:-:|:-:| | <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> | 1. Both do. 2. Only I is zero 3. Only II is zero 4. Neither is zero 5. ??? Note: * CORRECT ANSWER: B * Think about dE/dx and dE/dy * Fall 2016: 7 [34] 13 43 3; (Asked them to consider dvx/dx and dvy/dy) 3 [90] 3 4 0

What is the divergence in the boxed region? <img src ="./images/pt_charge_red_box.png" align="right" style="width: 400px";/> 1. Zero 2. Not zero 3. ??? Note: * CORRECT ANSWER: A * Lines in; lines out - harder to see dE/dx and dE/dy * One of those curious ones where the 2D picture might get in the way; think 3D

**Activity:** For a the electric field of a point charge, $\mathbf{E}(\mathbf{r}) = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r^2}\hat{r}$, compute $\nabla \cdot \mathbf{E}$. *Hint: The front fly leaf of Griffiths suggests that the we take:* $$\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2 E_r\right)$$ Note: * You get zero! Motivates delta function

Remember this? <img src ="./images/pt_charge_red_box.png" align="center" style="width: 400px";/>

What is the value of: $$\int_{-\infty}^{\infty} x^2 \delta(x-2)dx$$ 1. 0 2. 2 3. 4 4. $\infty$ 5. Something else Note: * CORRECT ANSWER: C

**Activity**: Compute the following integrals. Note anything special you had to do. * Row 1-2: $\int_{-\infty}^{\infty} xe^x \delta(x-1)dx$ * Row 3-4: $\int_{\infty}^{-\infty} \log(x) \delta(x-2)dx$ * Row 5-6: $\int_{-\infty}^{0} xe^x \delta(x-1)dx$ * Row 6+: $\int_{-\infty}^{\infty} (x+1)^2 \delta(4x)dx$ Note: * Give them 2-3 minutes to work on it and ask for what they did.