Begin2.DVI

Definition: (Equipotential curves) If F
 =
F(x,y )is a

given conservative vector field with potential φ(x,y ),then

the family of curves φ(x,y ) = care called equipotential

curves.

By selecting a constant value cand graphing the equipotential curves

φ=c, φ =c+ 1 , φ =c+ 2,... ,

one can determine by the spacing of these curves an estimate of the field intensity

in a given region.

The equipotential family of curves φ(x, y ) = csatisfies the differential equation

dφ =∂φ

∂x

dx +∂φ

∂y

dy = 0

or M(x, y )dx +N(x, y )dy = 0.

(9 .65)

If this differential equation is exact, then it can be expressed in the form

dφ =∂φ

∂x

dx +∂φ

∂y

dy =M(x, y )dx +N(x, y )dy

and using the figure 8-10, its solution may be expressed as a line integral in either

of the forms

φ(x, y ) =

∫x

x 0

M(x, y 0 )dx +

∫y

y 0

N(x, y )dy =c

or φ(x, y ) =

∫x

x 0

M(x, y )dx +

∫y

y 0

N(x 0 , y )dy =c

(9 .66)

depending upon the straight-line path of integration from P 0 to P. At any point (x, y ),

except singular points, where Mand Nare undefined, there is a tangent vector and

a normal vector to the point (x, y )on the curve φ(x, y ) = c. The vector F
=F
(x, y )lies

in the direction of the normal to the curve since grad φ=F
is a vector normal to

φ(x, y ) = cat the point (x, y )

Field Lines and Orthogonal Trajectories

Field lines are lines or curves such that at each point on these curves the direction

of the tangent vector to the curve is the same as the direction of the vector field at

that point. An orthogonal trajectory of a family of plane curves is a curve which

intersects every member of the family at right angles. The set of all curves which

intersect every member of φ(x, y )orthogonally are called the orthogonal trajectories