Illustration 25.2: Work and Equipotential Surfaces

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The animation shows the equipotential contours around a charge distribution.  The bar shows the work done to move the red test charge (position is given in meters, electric potential is given in volts and work is given in microjoules).  Equipotential surfaces are simply surfaces (in this two-dimensional representation they are lines) of constant electric potential.  Restart.

Equipotential contours are the same as a topographic contour map you might see for mountains (as below).  The contours are equally spaced so that each contour represents a given change in voltage (topographic maps have contours equally spaced for certain heights).  What is the difference in voltage between contours on the equipotential surface in the animation?

A mountainous region (left) and its contour map (right).
Image credits: United States Geological Survey: http://interactive2.usgs.gov/learningweb/teachers/mapsshow_act4.htm

• What is the work done on the test charge (by you) as you move it along the equipotential curve it started on? 
• What is the work done (by you) as you move it toward another charge? 
• The test charge is positive.  If the work done by you is positive as you move it toward a charge, what sign does that charge have?

The change in potential energy is proportional to the change in electric potential (with the charge being the proportionality constant).  If the change in electric potential is zero, then so is the change in potential energy, and therefore the work done is zero.  As you move a positive charge toward another positive charge, you must do a positive amount of work (the charges repel and you must counteract this).  As you move a positive charge toward a negative charge, you must do a negative amount of work (the charges attract and you must counteract this).

The electric field at any point is perpendicular to the equipotential contour.  You can see this by looking at the force vector on the test charge as you move the charge around in this potential field.  The direction of the electric field corresponds to the direction of steepest slope on a topographical map.  If this were a topographical map, this would be a map of three steep mountains and a steep valley.  How many positive and negative charges are on this electrostatic contour map?  Note that, since the electric field is perpendicular to the equipotential lines, if we move along an equipotential, no work is done (the electric force and the displacement are perpendicular).

Illustration authored by Anne J. Cox.
© 2004 by Prentice-Hall, Inc. A Pearson Company