Visualization of Potentials

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Section 4.3 Visualization of Potentials

In introductory physics, you may have learned how to find electrostatic potentials due to several discrete sources by first finding the electric field. However, it is also possible, and often useful, to find electrostatic potentials directly.

Activity 4.3.1. Drawing Equipotentials.

Using only what you know about the relationship of charges to electrostatic potentials, namely:

\begin{equation*} V=\frac{1}{4\pi\epsilon_0} \, \frac{Q}{r} \end{equation*}

and the superposition principle, sketch the equipotential surfaces for each of the following static charge configurations:

Four positive charges arranged in a square.
Two positive charges and two negative charges arranged in a square, with like charges diagonally opposite each other.
A line segment with constant charge density.
A circular loop with constant charge density.

Hint.

Some things that you might have needed to pay attention to are:

The electrostatic potential is a scalar field, not a vector field, i.e. it is a number at every point in space, not a vector.
The electrostatic potential at a given point due to several discrete charges is the scalar sum of the potentials due to the separate charges.
Pay attention to how the shape of the equipotential surfaces is related to the “shape” of the charge distribution.
It is easiest to sketch the potential close to the charges or far away. Pay some attention to intermediate points.
Pay attention to the line spacing. Should the lines be closer together in some places than in others? Why?
These examples are inherently 3-dimensional. For most people, drawing in three dimensions is challenging.
The potential is typically not zero at a point where the net electric field or the net force on a test charge is zero. Don’t claim that the potentials “cancel” just because the forces do.