1. Objectives

  • Represent {linear, surface, and volume} charge densities.

  • Calculate the electric field near a one-dimensional region having a charge density using integrals.

  • Use double integrals to calculate the electric field near a two-dimensional region having a certain charge density.

2. \(\mathbf{\hat{a}_R}\) vector direction

\[\hat{a}_R = \frac{\vec{r}_{here} - \vec{r}_{there}}{\left| \vec{r}_{here} - \vec{r}_{there} \right|}\]

This results in a relative position vector that points towards you from the other position. For example, you are at the origin and the other thing is at +10x, the \(\hat{a}_R\) sign is negative. See this in the “\(-\vec{r}_{there}\)” term.

3. First “useful integral”

After setup and some reduction in Example 4.1, you arrive a form that shows up on the Table of Useful Integrals:

\[\int{\frac{1}{\left(r^2 + z^2\right)^{3/2}} \mathop{}\!\mathrm{d}z} = \frac{z}{r^2 \sqrt{r^2 + z^2}} + C \label{zr2}\]
Compute this integral without using the table. Show your work and submit for an extra point.

4. All (correct) paths lead to the same answer

When I solved Example 4.3 before watching the video,[1], I took a different approach to setting up the integral.

These are posted to Canvas and available only after you submit your examples.

  • It only used the Pythagorean theorem and no trigonometry functions.

  • Found ρL by a side integral around the circle where the result was known to be Q.

  • A small hangup on the first pass was using \(\mathrm{d}l = \mathrm{d}\phi\) instead of the correct differential path length \(\mathrm{d}l = a\,\mathrm{d}\phi\).

  • My hand calculations used angle symbol \(\theta\) instead of \(\phi\).

Taking a slightly different approach to your solution that Prof. Tougaw does in the video is a good sign that you are actually solving the problem, as opposed to merely zombie-following.

Quizzes and especially exams are designed to be successfully completed by humans only. They are designed such that a zombie will typically score worse than random guessing.

You Have Been Warned [2]

5. Summary

The examples worked through common cases of charge distributions, with lots of symmetry. These geometries and partner solutions form the basis of many other geometrical variants by simply breaking down the problem into a superposition of these forms.

basis

As in basis functions or kernel (functions) that allow you to express a solution in terms of linear combinations of these special functions. Think of specifying any point in 3D space by combinations of the \(\hat{a_x}\), \(\hat{a_z}\), and \(\hat{a_z}\) unit vectors.

  • Beside each of these results, draw a sketch of the geometry that it represents.

6. Challenge 04

challenge04

See the note “Compute this integral…​” under equation \(\eqref{zr2}\) in § 3, “First “useful integral””.

“Clearly” means that the reader doesn’t need to open the web page in order to figure out what the solution is about. Provide a little context to your description and work showing why this is a “useful integral.”

1. Show me yours and I’ll show you mine for these exercise solutions.
2. Difference between Caution and Warning