Electric field at point Gauss law

[Pual Black]

Homework Statement

We won't to find out the electric field at a point p due to a point charge q placed at o as shown in the figure
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Consider a sphere of radius r passing through the point p. Let the electric field at p be E then by Gauss law

my problem is how we get the last step? I don't understand how he integrate it.

The Attempt at a Solution

i tried it and found a solution but not sure if its correct although i got the same result

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sorry that i don't used LATEX but I'm not used with it and i can't write the whole question with LATEX

 

[Doc Al]

You need to learn the rules for integration (and differentiation) of $x^n$. Try this: Power Rule

 

[BvU]

Looks as if the power rule is known, but the concept of integration causes difficulties: substitution $x=r^{-3}$ is carried out ok initially (*), but the much better (or rather, the 'right') substitution $x = r^{-2}$, leading to $dx =-2\; r^{-3}dr$ and hence $\int r^{-3} dr = -{1\over2}\int dx$ is overseen.

(*) Later on, the swap $\int {x\over r^{-4}}\; dx = {1\over r^{-4}} \int x\; dx$ is unforgivable: r is not a constant but depends on x !

 

[Pual Black]

If r is not constant and depends on x. Therefore i can't put the $r^4$ outside the integration. Right??

 

[BvU]

Indeed you can not !

 

[Pual Black]

Ok so how to get this solution?
Any idea??
I want the the same result $E=\frac{q}{4 \pi \epsilon r^2}$

 

[BvU]

OK, we have to make a step back. Your "Consider a sphere of radius r passing through the point p. Let the electric field at p be E then by Gauss law " continues with a picture with formulas that seem to be out of context: at P there is no charge, so div E is zero there !

Where does this snippet come from? It also integrates from $r=-\infty$ to $r=\infty$ which is very weird, since r can't be negative. Then it equates that to $2\int_0^\infty\$, then inserts a $- {1\over 3 }$ from nowhere (re-check the power rule)!

So, stepping back and trying to find the correct path:

Gauss law replaces a surface integral of E by a volume integral of rho. Indeed $\nabla \vec E = {\rho\over \epsilon_0}$, but $\rho \ne q\big /{4\over 3}\pi r^3\$ ! That is $\rho$ for a uniformly charged sphere with radius r !

In this simple case the volume integral of rho simply yields q. Frome there use rotation symmetry and gauss law to get to E.


Another thing: note that div E is not $dE\over dr$ !

Divergence in spherical coordinates tells us that

for rotational symmetry $\nabla\cdot\vec E = {1\over r^2}\; {\partial (r^2E_r)\over \partial r}$

(

as you can check: div E = 0 for the desired result for E

and conversely: with rotational symmetry, Er² is constant outside the area where charge is present.​

).

[edit] forgot the divergence link, inserted it.

 

[Doc Al]

Ok so how to get this solution?
Any idea??

Sure. Learn how to integrate $x^n$; in particular, $\frac{1}{x^3} = x^{-3}$. (No fancy substitutions needed.)

See the link I gave about the power rule.

 

[Pual Black]

Sure. Learn how to integrate $x^n$; in particular, $\frac{1}{x^3} = x^{-3}$. (No fancy substitutions needed.)

See the link I gave about the power rule.

I know how to use the power rule.
If you see my solution ;) @BvU
Thank you for your help. Looks like this question is a mess. The whole question is wrong
I will now use your information and maybe i will get a right solution.

 

[Doc Al]

I know how to use the power rule.
If you see my solution

Then why did you say the following?

my problem is how we get the last step? I don't understand how he integrate it.

The last step was integrating 1/x³.

I guess I missed something. (Sorry about that!)

 

[Pual Black]

Yes that's right but if you try to integrate the last step you will not get the same result. Therefore i said that i don't understand it. But as BvU told me the whole question has mistakes.

 

[BvU]

So is the picture with the formulas from a different exercise ? Or did you do a part of the solution in TeX and then switch to handwritten ?

 

[Pual Black]

No the two pictures ( figure and Tex ) are from exercise. Just the handwriting is mine. But the exercise it is not from one source cause the exercise has the soltuion from a screenshot. It has text and image ( the soultion and the figure ) are screenshots.

 

[BvU]

So, do we have the "all clear"now, or is there still some thing unfinished ?

 

[Pual Black]

Thank you very much. Its clear now