1. Drawing equipotentials. Two diagrams, each representing a force field in the
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Question
1. Drawing equipotentials. Two diagrams, each representing a force field in the x-y plane, are shown below (a) (2 points) For each case, is it possible to draw a self-consistent set of equipotential contour lines? If not, why not? (b) (3 points) If it is possible, draw a representative set of equipotential lines separated by the same constant value of U, including: (1) the correct shape of the lines, (2) the correct relative spacing of the lines, and (3) indicating which lines correspond to the highest and lowest potential energy. (You can draw right on the HW printout and staple it to your HW submission.)Explanation / Answer
A. The force field is given by the divergence of the potential field. Equipotential lines are lines that indicate points of equal potential and are consequently perpendicular to the field lines.
In the first case, the force field lines are forming loops. It is not possible to obtain a consistent set of equipotential lines. To see why this is, think of it this way. The lines perpendicular to the electric field are straight lines radiating from the center. But if we say one of those is a line of highest potential, and progressively follow the force field towards it to identify lines having lower potential than it, we eventually end up with the same line meaning that the points of the line have lower potential than themselves.. This is a contradiction.
It is possible to draw equipotential lines in the second case. They are concentric circles.
B. In the diagram, draw equipotential lines as circles centered at the point where all the field lines are pointing. The space between consequetive circles should increase as you move further away from the center. The circles closest to the center have highest potential energy and those far away have lowest.
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