Compute: $$\int_{-\infty}^{\infty} x^2\delta(3x+5)dx$$ 1. $25/3$ 2. $-5/3$ 3. $25/27$ 4. $25/9$ 5. Something else Note: * CORRECT ANSWER: C * Use a u substitution.

A point charge ($q$) is located at position $\mathbf{R}$, as shown. What is $\rho(\mathbf{r})$, the charge density in all space? <img src ="./images/pt_charge_at_R.png" align="right" style="width: 300px";/> 1. $\rho(\mathbf{r}) = q\delta^3(\mathbf{R})$ 2. $\rho(\mathbf{r}) = q\delta^3(\mathbf{r})$ 3. $\rho(\mathbf{r}) = q\delta^3(\mathbf{R}-\mathbf{r})$ 4. $\rho(\mathbf{r}) = q\delta^3(\mathbf{r}-\mathbf{R})$ 5. Something else?? Note: * CORRECT ANSWER: E * This one is a curious one because a delta function is always positive, both C and D are correct. * Expect most everyone to pick C

What are the units of $\delta (x)$ if $x$ is measured in meters? 1. $\delta(x)$ is dimension less (‘no units’) 2. [$\mathrm{m}$]: Unit of length 3. [$\mathrm{m}^2$]: Unit of length squared 4. [$\mathrm{m}^{-1}$]: 1 / (unit of length) 5. [$\mathrm{m}^{-2}$]: 1 / (unit of length squared) Note: * CORRECT ANSWER: D * Think about what the integral must produce.

What are the units of $\delta^3(\mathbf{r})$ if the components of $\mathbf{r}$ are measured in meters? 1. [$\mathrm{m}$]: Unit of length 2. [$\mathrm{m}^2$]: Unit of length squared 3. [$\mathrm{m}^{-1}$]: 1 / (unit of length) 4. [$\mathrm{m}^{-2}$]: 1 / (unit of length squared) 5. None of these. Note: * CORRECT ANSWER: E * Should be m^-3

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 * Just a check back in.