# Gauss' Law: Difficult Problem electric field etc

1. Feb 2, 2010

### Puchinita5

1. The problem statement, all variables and given/known data
A region in space contains a total positive charge Q that is distributed spherically such that the volume charge density ρ(r) is given by

ρ(r) = 3αr/(2R) for r is less than or equal to R/2
ρ(r) = α[1- (r/R)^2] for R/2 is less than or qual to r which is less than or equal to R
ρ(r)= 0 for r is greater than or equal to R

Here α is a positive constant having units of C/m^3

Using Gauss's law, derive an expression for the magnitude of the electric field as a function of r for R/2 is less than or equal to r which is less than or equal to R. Express your answers in terms of the total charge Q .

Express your answer in terms of variables r, R , Q and appropriate constants.

2. Relevant equations
This was a multi-step problem where I already had to figure out a couple things.

I have that

α= (480Q)/(233pi R^3)

I also have an expression for the magnitude of the electric field as a function of r for r is less than or equal to R/2.

E=(180Qr^2)/(233R^4*pi*ε) where ε is permittivity of space.

3. The attempt at a solution

see attached image. Hope you can follow everything!!! since it was a multi step problem, i highlight parts where i arrived at an answer. all my answers were correct except for the very last spot where i highlight a ridiculously long answer which is the one that i got wrong. If you can tell me what i did wrong, that would be great! maybe its a matter of simplifying???

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