Gauss's law -- Integral form problem

[jerryfelix30]

Homework Statement

The effective charge density of the electron cloud in a hydrogen atom in its quantum mechanical ground state turns out to be given by pnot(e^-(r/rnot)), where pnot is a negative constant (the clouds charge density at r=0) and rnot is a constant (rnot=0.025nm). Use gauss's law in integral form to calculate directly how E varies with r inside the electron cloud. Remember that there is a proton at r=0! Express your result in terms of the protons charge q.

Relevant Equations

Gauss's law= E dot dA=q(enclosed)/epsilon not
Q enclosed is the net charge enclosed in the shape and epsilonnot is the permittivity constant

Problem Statement: The effective charge density of the electron cloud in a hydrogen atom in its quantum mechanical ground state turns out to be given by pnot(e^-(r/rnot)), where pnot is a negative constant (the clouds charge density at r=0) and rnot is a constant (rnot=0.025nm). Use gauss's law in integral form to calculate directly how E varies with r inside the electron cloud. Remember that there is a proton at r=0! Express your result in terms of the protons charge q.
Relevant Equations: Gauss's law= E dot dA=q(enclosed)/epsilon not
Q enclosed is the net charge enclosed in the shape and epsilonnot is the permittivity constant

The shape is a sphere so area is 4pi r^2
Ex4pir^2=q/epsilonnot
E=q/4pir^2(epsilonnot)

 

[kuruman]

You need to rethink what $q_{enclosed}$ is if your Gaussian surface is inside the charge distribution. If you have a shell of radius $r$ inside the cloud, what fraction of the electron charge $e$ is enclosed by this shell? Hint: An integral is required.

 

[jerryfelix30]

You need to rethink what $q_{enclosed}$ is if your Gaussian surface is inside the charge distribution. If you have a shell of radius $r$ inside the cloud, what fraction of the electron charge $e$ is enclosed by this shell? Hint: An integral is required.

So you have to take the integral from 0 to r for the charge density with dq?

 

[kuruman]

So you have to take the integral from 0 to r for the charge density with dq?

Yes.