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