Inside this resistor setup, what can you conclude about the current density $\mathbf{J}$ near the side walls (in steady state)? <img src="./images/shaped_resistor.png" align="center" style="width: 600px";/> 1. Must be exactly parallel to the wall 2. Must be exactly perpendicular to the wall 3. Could have a mix of parallel and perp components 4. No obvious way to decide!? Note: * Correct Answer: A (otherwise current leaks out or there's accumulation!)
**Activity:** Consider two spheres (radii $a$ and $b$ with $b$>$a$) that are constructed so that the larger one surrounds the smaller one. Between them is a material with conductivity $\sigma$. A potential difference of $V$ is maintained between them with the inner sphere at higher potential. * What is the current $I$ flowing between the spheres in terms of the known variables? * How does your result relate to Ohm's Law? Hint: Assume a uniform charge $+Q$ distributed over the inner sphere and use Gauss' Law to find $\mathbf{E}$.
Recall the machined copper from last class, with steady current flowing left to right through it <img src="./images/machined_copper_2.png" align="center" style="width: 600px";/> In the "necking down region" (somewhere in a small-ish region around the head of the arrow), do you think 1. $\nabla \cdot \mathbf{E} = 0$ 2. $\nabla \cdot \mathbf{E} \neq 0$ Note: * Correct Answer: A
Recall the machined copper from last class, with steady current flowing left to right through it <img src="./images/machined_copper_2.png" align="center" style="width: 600px";/> In steady state, do you expect there will be any surface charge accumulated anywhere on the walls of the conductor? 1. Yes 2. No Note: * Correct Answer: A