2 Description of field lines

2.2.3 Equipotentials

Deducing the equations

The equation V(x,y) = is already in non-local form and will have as solution a curve in the xy plane (the intersection of a surface with the xy plane). The corresponding differential equations is:
or, remembering that the partial derivatives of potential V are the components of vector E:

and simply means that the normal to equipotentials in any point is the electric field in that point.

Analysis

Equipotentials are closed paths; some equipotentials close around one charge, some around more charges, and some enclose no charge at all (e.g.: four equal charges in the corners of a square). This can be a criterion for distinguishing families of equipotentials.

Given a point in the plane or a value of the potential V, it is difficult to predict the behavior of the corresponding equipotentials. However, the separation line between different families can be found using differential geometry considerations. Equipotentials are plane curves; the separation lines are given by those curves that have singular points (if an equipotential has no singular point, it is a "regular" one, not a limiting curve). The singular points of a curve are those for which all partial derivatives become zero simultaneously; but, since the partial derivatives of the potential V are the components of the electric field E, singular points are those for which E=0:

All we have to do is

• find the singular points of the field
• compute the potential V of those points
• trace the equipotentials corresponding to the computed V

Case study

a) charges of same sign and different magnitudes: q1 > 0, q2 > 0, |q1| < |q2|.

The singular point has been found to line on the horizontal axis at

The potential at that point (the separation potential) is

The equipotential corresponding to Vs consists of two curves intersecting at (xs , y). For lower potentials, the equipotentials enclose both charges; for higher potentials, the equipotentials consist of isolated curves. Equipotentials that encircle Vs tend to a round shape, as if the system of charges were acting as a single charge. Thus, Vs can be used as a threshold when approximating the system with a single charge.

b) charges of opposite signs and different magnitudes: q1 > 0, q2 < 0, |q1| < |q2|

The singular point has

The potential at that point (the separation potential) is

The image differs from the previous case; one has to consider the "zero" equipotential V=0, which turns out to be a circle. Then

• Positive values correspond to equipotentials that enclose the positive charge and are inside the "zero" equipotential.
• Negative potentials higher than Vs give birth to
• equipotentials that enclose both charges and decay progressively to circles and
• equipotentials that surround the positive charge and are larger than the "zero" equipotential
• Negative potentials lower than Vs are only matched by equipotentials that surround the negative charge

c) charges of same sign and same magnitudes, q1 = q2 = q > 0

This is just a particular case of (a). The separation potential is

d) charges of opposite signs and same magnitudes q1 = q > 0, q2 = -q < 0

This is the electric dipole case. There is a fundamental difference from case (b) : there is no singular point and no separation potential. The dipole can not be approximated by a point charge, no matter how far we are, because no equipotential surrounds both charges.

The "zero" equipotential (the vertical axis) is an axis of (geometric) symmetry for the equipotentials. On the left, equipotentials correspond to positive values; on the right, to negative potentials.