How Does Gauss's Law Apply to Nonuniform Spherical Charge Distributions?

[kbwelch17]

Homework Statement

A nonuniform, but spherically symmetric, distribution of charge has a charge density ρ(r) given as follows:
ρ(r) = ρ0(1-r/R) for r≤R
ρ(r) = 0 for r≥R

where ρ0 = 3Q/∏R^3 is a positive constant.

(a.) Show that the total charge contained in the charge distribution is Q.
(b.) Show that the electric field in the region r≥R is identical to that produced by a point charge Q at r = 0.
(c.) Obtain an expression for the electric field in the region r≤R.
(d) Graph the electric field magnitude E as a function of r.
(e.) Find the value of r at which the electric field is maximum, and find the value of that maximum field.

Homework Equations

Gauss' Law : Flux = ∫E dot dA = Qencl/ε 0
Electric Field of a point charge: E = k*(q/r^2)

The Attempt at a Solution

I am really just having trouble setting up the integral. You would be integrating the electric field over a sphere, so you would need to choose dA (is this a slice of the area of the sphere?) and a formula to find electric field for a given r. In order to find the latter you would need to use Gauss' Law in which case a give r would give the electric field

ρ(r) = q /A1 (is this the area or the volume of the sphere in question?)
q = ρ(r)*A1

E*A2 (what area is this?)= q/ε0
E = ρ(r)*A1 / A2*ε0

There are a few things I am confused about in here so please feel free to ask me to clarify anything. I need to know if I am on the right track here.

Thanks!

 

[vela]

Homework Statement

A nonuniform, but spherically symmetric, distribution of charge has a charge density ρ(r) given as follows:
ρ(r) = ρ0(1-r/R) for r≤R
ρ(r) = 0 for r≥R

where ρ0 = 3Q/∏R^3 is a positive constant.

(a.) Show that the total charge contained in the charge distribution is Q.
(b.) Show that the electric field in the region r≥R is identical to that produced by a point charge Q at r = 0.
(c.) Obtain an expression for the electric field in the region r≤R.
(d) Graph the electric field magnitude E as a function of r.
(e.) Find the value of r at which the electric field is maximum, and find the value of that maximum field.

Homework Equations

Gauss' Law : Flux = ∫E dot dA = Qencl/ε 0
Electric Field of a point charge: E = k*(q/r^2)

The Attempt at a Solution

I am really just having trouble setting up the integral. You would be integrating the electric field over a sphere, so you would need to choose dA (is this a slice of the area of the sphere?) and a formula to find electric field for a given r. In order to find the latter you would need to use Gauss' Law in which case a give r would give the electric field

ρ(r) = q /A1 (is this the area or the volume of the sphere in question?)
q = ρ(r)*A1

E*A2 (what area is this?)= q/ε0
E = ρ(r)*A1 / A2*ε0

There are a few things I am confused about in here so please feel free to ask me to clarify anything. I need to know if I am on the right track here.

Thanks!

For part (a), you want to use $dq = \rho\,dv$ and integrate over the entire sphere to find the total charge. You don't use Gauss's law here.