Potential and Lines of Force for Two Parallel Infinite Line Charges
The electrostatic potential in an \[Hyphen] plane for an infinite line charge in the direction with linear density is given by[more]
We use Gaussian units for compactness. The zero of potential is evidently the value on the circle .
For two parallel line charges, with linear densities and , intersecting the plane at and , respectively, the potential function generalizes to
For selected values of , and , selecting "contour plot" shows the equipotentials of . For , the equipotentials have the form of Cassini ovals. Also shown as green contours are the orthogonal trajectories , which represent the electrostatic lines of force. These are given by
as derived in the Details below.
A 3D plot of the potential contours is also available. Click the checkbox to display, for purposes of comparison, the analogous equipotentials and lines of force for two point charges  and replacing the line charges.[less]
The orthogonal networks of equipotentials and lines of force must satisfy the equation
or, more explicitly,
This is analogous to the mappings of the real and imaginary parts of a complex function. For the problem of parallel line charges, consider the complex function
where . Working out the real and imaginary parts of , we obtain the functions and given in the caption.