Potential of N Cylindrical Conductors of Infinite Length

[Zap]

The electric field of an infinite conductor of net charge Q along the x-y plane is easily found using Gauss's Law:

$$ \vec E(x, y) = \frac {\lambda} {2\pi \epsilon}\frac {[(x-x_c)\hat x + (y-y_c)\hat y]} {[(x - x_c)^2 + (y - y_c)^2]^3}, $$

where $x_c$ and $y_c$ mark the location of the center of the cylinder on the $x$ and $y$ axes respectively and $\lambda$ is the linear charge density.

In the electrostatic case, the potential can be found by solving $\vec E = - \nabla V$ by the method of separation of variables from the radius of the cylinder $R$ to any point along the x-y plane $r = \sqrt {(x-x_c)^2 + (y-y_c)^2}$:

$$\int_R^r dV = -\frac{\lambda}{2\pi \epsilon} \int_R^r \vec E \cdot d\vec r .$$

This gives

$$ V(x, y) = V(R) - \frac{\lambda}{2\pi \epsilon} \ln (\frac{R}{\sqrt{(x-x_c)^2 + (y-y_c)^2}}),$$

in which $V(R)$ is the potential of the cylinder.

This seems to give a good result for a single cylinder. However, when using the superposition principle for $N$ parallel cylinders of equal radius $R$ and linear charge density $\lambda$, the following is found:

$$ V(x, y) = \sum_{i=1}^N V_i(R) - \frac{\lambda}{2\pi \epsilon} \sum_{i=1}^N \ln (\frac{R}{\sqrt{(x-x_i)^2 + (y-y_i)^2}}). $$

A contour plot of this result gives N cylinders, but they are all at the same potential $\sum_{i=1}^N V_i(R)$.

My question is, is there anyway to define $N$ infinite and parallel cylinders which are at different voltages?

I am trying to create an approximate model of a grid made of around 24 parallel wires, which are not at the same potential.

 

[BvU]

By the same procedure, only you leave $\lambda_i$ inside the summation.

I see a parallel with drift chamber field shaping wires, which google; e.g. https://www.researchgate.net/figure/Layout-of-a-drift-chamber-cell-where-a-sensing-wire-is-placed-in-the-center-of-an_fig2_40618974

 

[Zap]

But the natural log makes it so that the potential on all the wires go to $\sum V_i(R)$, regardless of what $\lambda_i$ is, and the potential far away goes to negative infinity.

I don't think it makes sense.

 

[BvU]

Can't have V = 0 at infinity for infinite wires. However far you go, you see the same line charge.

Has to do with https://www.iopb.res.in/~somen/Courses/QM2016/RG_electro.pdf -- too complicated for me.

 

[Zap]

Right. I think that is why this doesn't appear to be a good model for this application.

The logarithm sets $\Delta V$ to zero on the surface of all the electrodes. The superposition principle doesn't work here, at least in the way I want it to, because we need the condition that voltage goes to zero at infinity and use that as the reference point for each electrode. Then, we can set the surface of each electrode to a different voltage and summing would produce the correct result. Here, it doesn't work, because each electrode's reference point is its own surface. Therefore, the voltage at the surface of all electrodes cannot be different, because the reference points must be the same.

I looked at a finite line of charge as another possibility. However, that one is not good either, because the radius of the wires is a parameter we are trying to investigate.

I think a ring of charge is probably the best approximation for the field along a two dimensional plane, but I am struggling with solving the integral.