1. Oct 27, 2009

### kliker

A region in space containing cargo which is distributed spherically so that the stocking load ρ is given by the equotations:

ρ = a for r<=R/2
ρ = 2a(1-r/R) for R/2<=r<=R
ρ=0 for r>=R

The total charge Q is 3*10^-17 C, the radius R of the overall distribution is 2*10^-14 m and a is a constant which has dimensions of C / m^3. Determine the constant a as
function of Q and R and the numerical value. b) Using the
Gauss's law find an expression for the measure of the electric field as a function of distance r
from the center of distribution. Do this separately for each of the three areas.
Make sure that your results agree with the borders of the areas. What fraction of
total load is contained within the region r<R/2?

i spent like 3 hours to solve this, but no success

2. Oct 27, 2009

### jdwood983

How is charge density, $$\rho$$, related to total charge $$Q$$ for a spherical charge? If you can find that relation, the first part is solved by inserting your values of $$\rho$$ in that relation.

What is Gauss's Law? More accurately, what is the equation for Gauss's Law? If you know the equation, you should be able to insert the charge density into this and find the answer.

3. Oct 27, 2009

### kliker

i know that ρ = Q/(4/3pir^3) right?

if i take r as R/2 and find ρ, how can i find a? i mean for R/2 we have two different equotations, then

if i take a gausian surface the Qencl would be Qencl = ρ * V where V = 4/3pir^3 then E would be E = Qencl/eo*Qencl

i really think it's something like this but im not 100% sure

ps: this is from the book "university physics by d young"

4. Oct 27, 2009

### jdwood983

You have three areas where the charge density exists:

$$\rho=\left[\begin{array}{ll}a &\,\,\,\,\,r<\frac{R}{2} \\ a\left(1-\frac{r}{R}\right)&\,\,\,\,\,\frac{R}{2}\leq r\leq R \\ 0&\,\,\,\,\,r>R\end{array}\right.$$

I would solve the middle one for $$a$$ in terms of $$\rho$$:

$$\rho=a\left(1-\frac{r}{R}\right)\rightarrow a=\frac{\rho}{1-\frac{r}{R}}$$

Then replace $$\rho$$ with your total charge that you have above. The electric field equation you have is wrong, you should have:

$$E_r=\frac{\rho}{\varepsilon_0}=\frac{Q_{enc}}{V\varepsilon_0}$$

5. Oct 27, 2009

### kliker

thank you again, this is actually what i did before, the answer in the book for a question is a = 8Q/5piR^3

but i keep finding a = 6Q/piR^3

Last edited: Oct 27, 2009
6. Oct 27, 2009

### gabbagabbahey

Why would you do that?

7. Oct 27, 2009

### kliker

hello, gabbagabbahey, what would you do in this case? i'm really stuck, this is like the last exercise i need to solve in order to finish my assignment

8. Oct 27, 2009

### gabbagabbahey

No, that is only when the charge is uniformly distributed over a sphere of radius 'r'...More generally, the volume charge density is defined according to the equation $dq=\rho dV$, where $dq$ is the amount of charge located in the infinitesimal volume element $dV$....to get the total charge of any distribution, you need to integrate over all space (or at least a volume that encloses the entire charge distribution):

$$Q=\int_{\text{all space}}\rho(\textbf{r})dV$$

9. Oct 27, 2009

### kliker

can you explain me what ρ(r) and "all space" are?

from the integral i found that Qw = Q*ln(V) where V = 4/3pir^3

10. Oct 27, 2009

### gabbagabbahey

Haven't you encountered volume integrals before? Have you taken 2nd year calculus?

11. Oct 27, 2009

### kliker

no i dont think so, im studying computer science 1 year, sorry for my bad english i'm from russia

12. Oct 27, 2009

### gabbagabbahey

You really need to have taken a multivariable calculus course to understand electrodynamics...what course is this problem for, and what textbook are you studying from? Are you familiar with spherical coordinates?

13. Oct 27, 2009

### kliker

it's for physics, this exercise (23.27) is from the book university physics by Hugh D. Young

where can i find these courses? i really need to understand and solve this by tomorrow

Thanks

14. Oct 27, 2009

### gabbagabbahey

That text has a few examples of volume integrals in spherical coordinates, I suggest you look at them.

15. Oct 27, 2009

### kliker

thank you for the answers, i ve studied everything in there

after some time i found this AWESOME guy http://www.youtube.com/watch?v=nvq_4OFp_IE&NR=1

explaining different examples, i watched all his videos, and i think i understand what i have to do now

if i find any problems i ll update this thread

thanks everyone

16. Oct 27, 2009

### jdwood983

I would've done that because the answer requested the electric field in the three different areas. I was using that as just a suggestion to solve the problem, but maybe I didn't say it was the complete answer but it is for sure a starting point.

Also, Young's book is an algebra based text (at least the copy I have is), so I'm assuming that he doesn't know anything about integrals, hence not suggestion using one.

17. Oct 27, 2009

### kliker

after some search

because the distribution is not done uniformly

dq = ρ * dv

Qencl = integral(ρ*dv)

Qencl = integral(ρ*4*pi*r^2)dr

what would ρ be here though? in the above video the man says it's k*r but shouldn't it be Q/(4/3*pi*r^3)?

im stuck again...

life can be sucky sometimes :(

18. Oct 27, 2009

### gabbagabbahey

You are given $\rho$ in your problem statement...

19. Oct 27, 2009

### kliker

yes i used it, but again i cant find the correct answer which is

a = 8Q/(5*pi*R^3)

20. Oct 27, 2009