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Learning Introductory Physics with Activities

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

Section 22.12 Application: Nonuniform Wire

Subsubsection Activities

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Activity 22.12.1. The Nonuniform Wire.

A wire of length \(2L\) lies along the \(x\)-axis and has its center at the origin. The non-uniform charge density of this wire is known to be
\begin{equation*} \lambda(x) = +\frac{Q_o}{L^2}|x| \end{equation*}
You are interested in finding the electric field created by this wire at a point along the \(y\)-axis.
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(a)

Draw a graph of the charge density. Use your graph to draw a physical picture of the wire showing the location of charges in the wire.
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(b)

Predict the direction of the electric field at the indicated point.
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(c)

Choose a small section of the wire and write a symbolic expression for the charge on that section \(dq\text{.}\) Label your figure with all relevant distances.
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(d)

Find the electric field created by your dq at a point along the \(y\)-axis.
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(e)

Set up an integral to find the electric field created by the entire wire at a point along the \(y\)-axis.
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Answer.
The electric field created by this wire at a point along the \(y\)-axis is
\begin{equation*} \vec{E} = \frac{2kQ_o}{L^2}\left(\frac{\sqrt{L^2 + y^2} - y}{\sqrt{L^2 + y^2}}\right)\hat{y} \end{equation*}
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(f)

Make sense of this answer using the following strategies:
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  1. Check that the units are correct for electric field!
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  2. Check that it points in the correct direction.
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  3. Evaluate the answer in the special case that \(y\) goes to \(\infty\text{.}\) (Remember to compare the actual answer to your expectation!)
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  4. Sketch a graph of the electric field along the \(y\)-axis (Desmos might help!). Does your sketch have the right qualitative behavior?
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