Voltage drop in an infinite wire

[YaroslavVB]

Suppose I have finite wire of known resistivity. I know voltage is 0 volts at x=-1 and x=1, and 1 volt at x=0. How do I find voltage at intermediate points?

 

[berkeman]

Suppose I have infinite wire of known resistance and inject current into position x=0. How do I find voltage distribution?

An infinite wire would have an infinite resistance. I think you mean a known resistivity, or resistance per unit length. If a current is induced in the wire, you just use Ohm's Law to calculate the voltage drop per unit length.

 

[YaroslavVB]

Yes, I probably mean resistivity, and know voltage instead of current, updated post with fixes

 

[K^2]

It's going to be 1V everywhere, and no current will flow, because the total resistance is still infinite.

 

[YaroslavVB]

OK, another update.

 

[berkeman]

OK, another update.

It's better if you just post an updated question in each of your replies, instead of editing the original post. It's confusing if you keep changing the original question.

To try to answer your question, since the current must be the same everywhere in the wire, the linear voltage drop will be the same everywhere. So if you have two points with known voltages, the voltage will vay linearly between those two points.

 

[K^2]

Yup. So V(x) = (1 - |x|) Volts.

In general:

$$I = \sigma E(x)$$

For all x, where sigma is conductivity = 1/resistivity, and

$$V(a,b) = \int_a^b dx E(x)$$

(Did I miss a minus sign somewhere? I feel like I did...)

 

[YaroslavVB]

I was trying to see if limiting distribution of a a symmetric random walk on R can be modeled as voltage, but now it doesn't seem there's a direct connection

 

[K^2]

No. No connection. But random walks are related to heat flow. You might want to look at that.

 

[YaroslavVB]

What's a good textbook for that?