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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

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

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

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.

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