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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 25.12 Challenge - Representing Potential

A*R*C*S 25.12.1. Infinite Wires.

Two infinite wires are located parallel to the y-axis. One wire has uniform charge density \(+\lambda\) and is located at \(x = +x_o\text{.}\) The other wire has uniform charge density \(-\lambda\) and is located at \(x = -x_o\text{.}\) The electric potential in the \(xy\)-plane for this system is given by:
\begin{equation*} V(x, y) = 2k \lambda \ln{|x + x_o |} - 2k \lambda \ln{|x-x_o |} = 2k \lambda \ln{| \frac{x + x_o }{x - x_o} |} \end{equation*}
Calculate the electric field at every point in the \(xy\)-plane.
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Tip: For your physical representation (part 1c), create an accurate equipotential graph in the \(xy\)-plane and the corresponding electric field vector map.
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Tip: For your symbolic sensemaking (part 3c), show that your electric field is what you expect from the superposition of two infinite line charges.
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Explanation 25.12.2. Exploring an Equipotential Graph.

An equipotential map with many concentric circles at the top and many ellipses at the bottom. A, B, and C lie on the same nearly circular contour near the top, while C, D, and E lie in a straight line from top to bottom.
Figure 25.12.1. Equipotential graph in the \(xy\)-plane.
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This problem is a modified Explanation Task with several specific prompts. Above is a graph of the equipotentials due to some unknown distribution of charges, with \(x\) and \(y\) measured in meters. The boxed numbers represent the values of the potential in volts.
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  1. Sketch vectors at the indicated points that represent the electric field \(\vec{E}\text{.}\) Both the direction and the magnitude of your vectors should be qualitatively correct. Describe how you determined your answers.
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  2. Explicitly rank the magnitudes of the electric field vectors you drew. Describe how you determined your answers.
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  3. Sketch a charge diagram, using \(+\) and \(-\) to indicate the locations where you expect there to be positive and negative charges, respectively. Justify your answers.
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  4. Imagine that you are a charged particle moving along the vertical line that passes through points C, D, and E (the \(y\)-axis).
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    1. Sketch a graph of the electric potential \(V\) as a function of \(y\text{.}\)
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    2. On a different set of axes, sketch the \(y\)-component of the electric field \(E_y\) as a function of \(y\text{.}\)
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    3. Discuss the relationship between your two graphs.
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Subsubsection Metacognitive Reflection

Activity 25.12.3. Options for Calculating the Electric Field.

You now have a few different ways to calculate the electric field from the electric potential. Describe how you can determine the electric field if you are given: 1) an equation for the potential everywhere in space, or 2) an equipotential map. Be sure to discuss how to determine the direction in both cases.
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