<img src ="./images/pt_charge_red_box.png" align="left" style="width: 300px";/> This picture represents the field lines of a single positive point charge. What is the divergence in the boxed region? What is the divergence of the whole field? 1. Boxed region is zero; whole field is zero 2. Boxed region is non-zero; whole field is zero 3. Boxed region is zero; whole field is non-zero 4. Boxed region is non-zero; whole field is non-zero 5. ??? Note: * CORRECT ANSWER: C
**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.
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.
We have shown twice that $\nabla \cdot \mathbf{E} = 0$ using what seem to be appropriate vector identities. But physically, $\nabla \cdot \mathbf{E} = \rho/ \varepsilon_0$. What is going on?! 1. We broke physics - let's call it a day 2. There's some trick to get out of this and that makes me uncomfortable 3. I can see what we need to do 4. ???