What Fraction of Total Charge Lies Between (R/2) and R?

[hitek131]

A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density rho (r) is given by:

rho (r)=3 * alpha * r /(2R) for r ≤ R / 2

rho (r)= alpha * [1-(r/R)^(2) ] for (R/2) ≤ r ≤ R

rho (r)=0 for r ≥ R
Here alpha is a positive constant having units of C/m^3What fraction of the total charge is contained within the region (R/2) ≤ r ≤ R?

If anyone can help me with this problem it would be greatly appreciated. I am getting lost in all the integration with everything. Thanks

 

[jtbell]

Please show us what you've done so far...

 

[hitek131]

This is what I got so far. It aint pretty.

The integral from (r/2) to r of:

Q/ (3R^3*pi/32)+(47R^3*pi/120) all this * (1 -(r/R)^2) * (4 * pi*r^2 dr)
all divided by 4 * pi *r^2 * eplison not.

 

[jtbell]

Please back up a few steps and explain how you got to that point.

 

[hitek131]

I am not really sure what I did. I used some integration formulas's, but I don't think they are right. Do you have a better starting way then what I did?

 

[hitek131]

For some reason, i don't know where to start off on this problem. There are so many specifications and I don't know if I'm doing the problem the right way.

 

[jtbell]

OK, start with Gauss's Law. What does it say? How can you relate it to the conditions of your problem?

 

[hitek131]

I know that Gauss's Law is the integral of vector E * vector dA

 

[hitek131]

I could use Gauss's Law and use the second condition from the specs in the problem. I could take the integral from (R/2) to R of the second condition.

 

[Astronuc]

Think about the electric field or rather electric flux and the 'enclosed' charge.

For example, reference - http://hyperphysics.phy-astr.gsu.edu/HBASE/electric/elesph.html#c1

 

[hitek131]

I found the integral. It is ( a (1- (r^2/R^2) * R)/2
Now I don't know how to use the Law to compute anything?

 

[jtbell]

I know that Gauss's Law is the integral of vector E * vector dA

That's the definition of the flux of $\vec E$ through a surface.

Look up Gauss's Law in your textbook. It says that the total flux of $\vec E$ through a closed surface equals the total charge enclosed inside that surface divided by $\epsilon_0$. Both of these can be written as integrals:

$$\int{\vec E \cdot d \vec A} = \frac{1}{\epsilon_0} \int {\rho dV}$$

where $\rho$ is of course the charge density.

 

[jtbell]

Bleah, the server must be overloaded. My equations haven't shown up in my preceding posting yet. [oops, I had to fix a bug... now they're OK]

Anyway, to use Gauss's Law, imagine a spherical surface of radius r (the radius that you want to find E at). You need to find (a) the flux of E through that suface, in terms of r and E and constants, and (b) the total charge enclosed inside the surface, using the densities that you've been given. Then substitute into Gauss's Law and solve for E.

Note that integrating the charge density between R/2 and R (as you proposed) gives you the total charge contained inside that spherical shell. The r that you want to find E at, is < R, so your imaginary spherical surface doesn't contain all of that charge.

Also, don't forget to include the charge that lies between radius 0 and R/2!

 

[hitek131]

I don't know how to relate electric flux to this problem.