satisfied by the equipotential surfaces is obtained by differentiating φ(x, y, z) = cto
obtain the exact differential
dφ =
∂φ
∂x dx +
∂φ
∂y dy +
∂φ
∂z = 0
or F 1 dx +F 2 dy +F 3 dz =F·dr = 0
(9 .82)
The solution of this differential equation may be obtained by line integration meth-
ods.
The field lines associated with the vector field F are those curves which are
everywhere tangent to the field vectors. The direction of the field lines are in direct
proportion to the components of F and thus the differential equation satisfied by the
field lines is dr =kF, where kis a proportionality constant. Equating like components
produces the equations
dx
F 1 (x, y, z )
= dy
F 2 (x, y, z )
= dz
F 3 (x, y, z )
=k (9 .83)
which is equivalent to the statement that dr ×F = 0 since dr has the same direction
as F . Another way of picturing this is to let r denote the position vector to a point
(x, y, z )on a field line. The differential element dr will then be in the direction of
the tangent to the field line which, by definition, is also in the same direction as F
at the common point (x, y, z).Thus, dr =kF , where kis a proportionality constant.
This equation can be written in the component form as
dr =dx ˆe 1 +dy ˆe 2 +dz ˆe 3 =k[F 1 (x, y, z)ˆe 1 +F 2 (x, y, z)ˆe 2 +F 3 (x, y, z )ˆe 3 ].
Equating like components produces the differential relation (9.83). Geometrically,
the field lines defined by equation (9.83) are orthogonal to the equipotential surfaces
defined by equation (9.82). That is, grad φis perpendicular to the tangent element
dr.
A solution of the differential system (9.83) consists of two independent relations
or integrals of the form
μ 1 (x, y, z) = c 1 and μ 2 (x, y, z ) = c 2 ,
which represents two families of surfaces having c 1 and c 2 as parameters. The field
lines are the curves of intersection of these two family of surfaces, and these curves
(field lines) are called a two-parameter family of curves, where the constants c 1 and
