1 Equipotentials

Thus far, vector fields and field lines were used to visualize a vector function. A third method used to visualize vector fields is the equipotential line.

A field line shows the direction of a vector function at all points along the line. An equipotential line is a line that is always perpendicular to a field line that passes through it. An equipotential line is created by

Selecting a starting point
Taking a small step perpendicular to the field line that passes through that point
Using the end of the step as a new starting point and then repeating 1.–3.

In the following figure, three steps are shown for an equipotential associated with the vector field $\mathbf{A}=\mathbf{\hat{r}}/r^2$ . At the starting point $(x,y)=(1,0)$ , the field line is horizontal, and so a small step is taken perpendicular to it, which is in the vertical direction. A the end of the step, the direction of the field line is different, and so the next step perpendicular to it is upwards and to the left. If we considered smaller steps and continued drawing the line, the result would be a circle centered on the origin.

The physical interpretation of an equipotential line is that if the field lines are associate with a force, the potential energy of an object that moves along the equipotential line will not change (the potential is equal at all points on the line).

Suppose there is a frictionless tube placed along the equipotential line. If a particle is given a kick so that its speed is initially $v_o$ , this speed will not change as it moves along the track. The two forces acting on the particle are the force associated with the field and the force of the track. Both of these forces are perpendicular to the direction of movement, and so the kinetic energy of the particle does not change. From the conservation of energy, $\Delta KE + \Delta PE=0$ , and if $\Delta KE=0$ , it follows that $\Delta PE=0$ , meaning that the potential of the particle is equal at all points along the equipotential line.

The mathematics of equipotentials will be covered later in the semester. At this point, you need only to understand the physical interpretation of equipotentials and how they can be drawn given a field line diagram.