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## Drawing Equipotential Surfaces for a Quadrupole: Instructor's Guide

### Main Ideas

Students are asked to draw equipotential surfaces due to several simple charge configurations starting with the formula for the electrostatic potential due to a point charge and the superposition principle.

1. They must clarify, in their own minds, the differences between the electric field $\Vec E$ and the electrostatic potential $V$. The electrostatic potential, $V$, is a scalar, not a vector.
2. What does the superposition principle mean?
3. The electrostatic potential, $V$, exists in three-dimensions, not just two.

Approximate Time: 45 minutes

Students are asked to draw lines of constant electrostatic potential due to a quadrupole distribution of charges. The prompt should ask them to use the formula for the potential due to a point charge and the superposition principle and not to use reasoning from other courses.

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### Activity: Introduction

Introduce the formula for the electrostatic potential due to a point charge: $$V(\Vec r)=\frac{1}{4\pi\epsilon_0} \frac{q}{|\Vec r - \Vec r'|}$$ and introduce the idea of the superposition principle.

### Activity: Student Conversations

Most students find this activity very challenging, even though these students had already done an activity in which they considered two point charges and were required to find the series expansion for the electric potential along an axis.

The types of things that were common for students to do:

• Explicitly think in terms of force and field vectors instead of scalar potentials. This resulted in a variety of conceptual misunderstandings, including thinking that the potential at the center of four positive charges would be zero. This is related to the misunderstanding that if the force on the particle is zero, than the potential must also be zero.
• Students very much want to rely on their intuition about field lines to lead to equipotential surfaces, even after the instructor specifically asked them to reason from the potentials equation, and not use field lines. Some students actively think about the field lines as they drew them, while others seem to draw the field lines as if they were purely memorized.
• Not considering the spacing of equipotential lines. Frequently students have equipotential lines more closely spaced farther away from the charges than close to the charges. When students draw equipotential lines perpedicular to field lines, they frequently seem to start at random locations and then just start drawing perpendicular lines. The resulting equipotential lines could be bunched up in one place and not in another, in a way that had no correlation to the rate the potential was changing.
• Students sometimes used mutually contradictory reasoning at different points. For example, they might correctly conclude that the potential midway between a positive and negative charge of equivalent magnitude would be zero, but then discuss forces and canceling vectors at some other point on their drawing.
• Students have a difficult time determining the steepness of the potential profile. Specifically, many students think that in the case of 4 positive charges, the potential profile is steeper in the middle of the charges than it is on the outside (presumably because the value of the potential is greater there). It is helpful to have students compare: (1) points equally distant from a single charge (one inside the distribution and one outside) and (2) two points where the potential is the same value but are located different distances from the source charges (where the potential is zero is a nice choice). Students can use these points to determine the “gradient” of the potential (and thereby, the relative spacing of the equipotential lines).
• For the case of the four positive charges, students aren't sure what's happening at the point in the center of the distribution. A few students seem to think that there is a dip there — and some will think the potential must be zero there. In particular, some students can't decide if a point on the edge of the square (equidistant from the two corners) has a larger potential than the point at the center. Students should be encouraged to use trig and algebra to determine which one is larger.
• Many students (and some instructors!) will want to argue that, since the potential due to a positive charge is positive, the potential at the center of four positive charges must be positive. But this argument is incorrect! This is a good time to remind students that we care about potential differences, and that the standard convention (in most cases) is to set the potential at infinity to zero. Only after making this conventional choice can one make the above positivity argument.
• A good discussion question is whether level curves can cross or intersect.

### Activity: Wrap-up

Emphasize what information about the equipotential surfaces comes from the formula $$V(\Vec r)=\frac{1}{4\pi\epsilon_0} \frac{q}{|\Vec r - \Vec r'|}$$ together with the superposition principle, and what information comes from other things that they may have learned in intro courses about the electric field $\Vec E$.

### Extensions

This activity is part of the sequence of activities addressing Representations of Scalar Fields in the context of electrostatics.

• Preceding activities:
• Electrostatic Potential due to a Point Charge: This small whiteboard question asks students to recall the electrostatic potential due to a point charge which results in discussions likely to include notation of the distance from th origin to a point charge.
• Follow-up activities:
• Visualizing Electrostatic Potentials: Students begin by brainstorming ways in which to represent three-dimensional scalar fields in two-dimensions and then use a Mathematica notebook to explore various representations for a distribution of point charges.
• Electric Potential Due to a Ring Mathematica Extension: This small group activity begins with students solving for the electrostatic potential due to a charged ring everywhere in space, an elliptic integral, and then use power series to approximate the potential at various locations in the scalar field. As an extension, students use a Mathematica notebook to visualize the electrostatic potential over all space in various representations.

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