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Learning Introductory Physics with Activities

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

Section 23.9 Using Symmetry

Subsubsection Activities

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Activity 23.9.1. Infinite Slab.

A charged wire with length \(L\) is oriented horizontally. The wire has a positive uniform charge density (the charge is spread out uniformly along the one-dimensional wire).
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Left, a 3D view of a rectangular slab that occupies the entire xy-plane, extending through a z-height of 2h. Right, an edge view of the yz-plane, showing a rectangular cross section of the slab, extending infinitely in the horizontal y direction and vertically from -h to h in the z direction. Point A lies above the slab on the left, point B lies above the slab on the right, point C lies in the slab on the right, and point D lies in the slab on the left, the four points forming a rectangle.
Figure 23.9.1. Two views of a charged slab extending infinitely in the x- and y-directions.
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(a)

Which direction should the electric field point?
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(b)

How does the magnitude of the electric field at A and B compare? How does the magnitude of the electric field at C and D compare?
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(c)

Which method should you use: Gauss’s Law or chop-multiply-add? How do you know?
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(d)

Use the following steps, first for \(z \gt h\) (outside), and then for \(z \lt h\) (inside).
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  1. Draw an appropriate Gaussian surface.
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  2. Determine the enclosed charge.
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  3. Write down the flux through each surface.
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  4. Use Gauss’s Law to determine the magnitude of the electric field. Do not forget your direction!
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Activity 23.9.2. Finite Slab.

Suppose you have a finite slab of thickness \(2h\text{,}\) length \(2a\text{,}\) and width \(2b\text{,}\) with a uniform volume charge density \(\rho\text{.}\) You want to determine the electric field everywhere in space.
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Left, a 3D view of a rectangular slab that has x-length 2a, y-width 2b, and z-height 2h. Right, an edge view of the yz-plane, showing a rectangular cross section of the slab, extending horizontally in the y direction from -b to b and vertically from -h to h in the z direction. Point A lies above the middle of the slab on the z-axis and point B lies horizontally to the right from point A.
Figure 23.9.2. Two views of a finite charged slab.
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(a)

True or False: The direction of the electric field at point A and B is the same.
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Answer.
False.
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(b)

Try setting up an integral using chop-multiply-add for no more than five minutes before you click on the hint.
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Hint.
Setting up this integral is extremely challenging, and evaluating it analytically is nearly impossible. Click the solution to see what it looks like!
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Solution.
Top, a 3D view of a rectangular slab, with a small cube representing the infinitesimal piece. Bottom, an edge view of the yz-plane, showing the small cube at position r prime and the position r where the field is to be evaluated.
Figure 23.9.3. Two views of an infinitesimal chunk of a finite charged slab.
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\begin{equation*} \vec{E}(x,y,z) = k\rho\int_{-h}^h \int_{-b}^b \int_{-a}^a \frac{\left(x - x'\right)\hat{x} + \left(y - y'\right)\hat{y} + \left(z - z'\right)\hat{z}}{\left(\left(x - x'\right)^2 + \left(y - y'\right)^2 + \left(z - z'\right)^2\right)^{3/2}}dx'dy'dz' \end{equation*}
Calculating the electric field for any given charge distribution is in general non-trivial. This is why you try to apply any symmetry arguments before diving into the chop-multiply-add method.
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