of the family. Let ψ(x, y ) = c∗ denote the family of orthogonal trajectories to the
family of equipotential curves φ(x, y ) = c. The family of curves ψ(x, y ) = c∗describes
the field lines associated with the vector field F . That is, every orthogonal trajectory
of the family of equipotential curves φ(x, y ) = chas a tangent vector which lies along
the same direction as the vector F (same direction as the normal to φ(x, y ) = c.) If
r =r (x, y )defines a field line, then dr points in the direction of vector field so that the
slope of the field lines are in direct proportion to the components of F. If dr =kF,
where kis some constant, then one can write dx ˆe 1 +dy ˆe 2 =k[M(x, y )ˆe 1 +N(x, y )ˆe 2 ]or
after equating like components
dx
M =
dy
N =k or −N dx +M dy = 0. (9 .67)
This gives the differential equation which defines the field lines. An equivalent
statement is that dr ×F = 0 ,where r is the position vector to a point on the field
line curve ψ(x, y ) = c.
Example 9-5. Show that the vector field
F=M(x, y )ˆe 1 +N(x, y )ˆe 2 =xˆe 1 +yeˆ 2
is conservative and sketch the equipotential curves and field lines associated with
this vector field.
Solution: The vector field is conservative, since curl F = 0 .If φ(x, y ) = cis a family
of equipotential curves, then dφ = grad φ·dr =F ·dr = 0 produces the differential
equation of the equipotential curves and one can write
dφ =M dx +N dy = 0 or dφ =x dx +y dy = 0.
By integrating this equation, there results the equipotential curves
φ(x, y ) =
x^2
2 +
y^2
2 =c,
which are circles centered at the origin.
If r is the position vector to a point on a field line, then dr is in the direction of
the tangent to the field line and must have the same direction as the vector field F
so that one can write dr =kF, where kis a proportionality constant. Equating like
components one then finds the differential equation describing the field lines as
dr =dx ˆe 1 +dy ˆe 2 =kF 1 ˆe 1 +kF 2 ˆe 2 or
dx
F 1 =
dy
F 2 =k or
dx
x =
dy
y =k.
