Electron flow over a conducting surface of variable resistivity

[fyzxfreak]

Hello fellow physics-people,

I was just thinking about the following setup:
We have a conducting surface (with smoothly varying resistivity) hooked up to some battery with the wires contacting the surface at two arbitrary points, A and B. How would we go about figuring out the current density?

Would the electrons flow in a simple curve from point A to B (to follow the path of least resistance) or would they spread out (as there might be a configuration in which parallel flow reduces mutual electric repulsion)?

Any ideas/suggestions? Thanks!

 

[Drakkith]

I believe that the current would flow through a large area of the conductor. But I'm not sure.

 

[fyzxfreak]

Is there a systematic method of approaching this problem? i.e. application of Maxwell's equations, etc.

=EDIT= There'd definitely be some variational calculus involved (for the different paths from point A to point B). Yeah... this problem does not seem particularly simple anymore, haha.

 

[Drakkith]

Is there a systematic method of approaching this problem? i.e. application of Maxwell's equations, etc.

I'm not knowledgeable enough to answer. I was basing my guess on knowing that the more conductor a current can flow through, the less the resistance overall. I would think this is balanced against the resistance caused by the longer path some of the current takes, resulting in an equilibrium somewhere. But all this is merely a guess.

 

[sardar]

I would approach this scenario by first considering the fundamental principles of electricity and conductors. In this case, the conducting surface with variable resistivity would act as a conductor, allowing the flow of electrons when a potential difference (provided by the battery) is applied.

The flow of electrons in a conductor is governed by Ohm's Law, which states that the current density is directly proportional to the electric field and inversely proportional to the resistivity. In this case, as the resistivity of the conducting surface is smoothly varying, the current density would also vary along the surface.

To determine the current density, we would need to calculate the electric field at each point on the surface. This can be done using the equation E = V/d, where E is the electric field, V is the potential difference, and d is the distance between the points A and B.

Now, to answer the question of whether the electrons would flow in a simple curve or spread out, it would depend on the specific configuration of the conducting surface and the potential difference applied. In general, electrons tend to follow the path of least resistance, so they would flow in a curve from point A to B. However, depending on the distribution of resistivity along the surface, there could be cases where parallel flow reduces mutual electric repulsion and the electrons spread out.

In conclusion, to determine the current density in this scenario, we would need to calculate the electric field at each point on the conducting surface and consider the distribution of resistivity. Further analysis and experimentation may be needed to fully understand the behavior of the electrons in this setup.