×

# Nonconducting nonuniform non-nonsymmetric cylinder

A long, nonconducting, solid cylinder of radius 4.0 cm has a nonuniform volume charge destiny $$p$$ that is a function of radial distance $$r$$ from the cylinder axis: $$p=Ar^2$$. For $$A=2.5 \mu /m^5$$, what is the magnitude of the electric field at (a) r = 3.0 cm and (b) r = 5.0 cm?

So, clearly, I overdid it. See #2 for a legible version of solution to (a). For me, apparently, that was step one. So now the question is, WHAT ABOUT REPULSION? See below, the concentric circles on top right:

For every differential excerpt $$dr$$, the enclosed charge $$q_enc$$ generates an electric field perpendicular to the circumference. This field points both ways - outward and inward. Now, don't inward electric field lines of $$r+dr$$ cancel those of $$r$$ and form a certain $$E_{net_{r}}$$?

At first, I've tried to formulate an integral based on $$E_r - E_{r+dr}$$. You can imagine how that went ($$\int ...(r+dr)dr$$...). Then, I thought instead I should do $$E_{net}=E_r-E_{R-r}$$. It led to the circled big formula , derived after an hour of globbling - above.

Something was wrong. This equation is clearly not elegant. And when physics throws you an elegant problem, you must get an elegant answer. Then I realized something... do you know what it is? It made me mad. I'll leave it to you to figure out ;)

NOTE Spoilers ahead! If you don't know the answer to above, give it a thought!

So, what if we DID have to resort to such measures? What if there was no internal symmetric cancellation? How would we resolve that issue? Say, for example, we have a big rectangle - and its charge density obeys the same pattern as that cylinder. If we place the rectangle on origin with length parallel to y axis, let $$r$$ be $$x$$. What would be its electric field lines (assume it's infinitely long)? NOW my issue is real.

2 years, 11 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$