Statement
An infinite line charge with charge density rL lies along the x axis, and the surface of zero potential passes through the point A in Cartesian coordinates. Find the potential at B.
System Parameters
Permittivity of free space:
Solution
Since the charge lies along the x axis, the x coordinates of the points P1 and P2 may be ignored. Thought of another way, the electric field is entirely in the y and z directions, so the work to move a charge in the x direction is zero. The radii used for calculation of the electric field are given by the y and z coordinates, as shown in the graph.
P.12 |
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P.13 Potential From a Line Charge |
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z axis
y axis
(To see the rest of the equations used to generate this graph, look to the right of this screen.)
The voltage is the line integral over the electric field. Pick a radial path to avoid parametrization of the line in Cartesian coordinates (see Problem 5.1). The reference point at which the potential is zero is necessary to define the integral, since potential describes a difference between two points, and not an absolute reference. The reference is, in fact, somewhat arbitrary.
Using the Simplify menu pick from the Symbolic menu (if you are using the Mathcad Engine, you won't have this option) produces
Evaluate this expression at the points A and B:
