Using Gauss's law to calculate magnitude of electric field

[Signifier]

I understand how to use Gauss's law to calculate the electric field at some point for say a sphere with charge distributed uniformly in it. I am bit confused, though, about calculating the electric field at some point for a non-uniform charge distribution.

For example, say that I have a spherically symmetric negative charge distribution around the origin of my reference frame, with charge density p(r) given by -0.5e^(-r), where r is the distance from the origin (IE, the radius of a given sphere) (so, p(0) = -0.5 and p(1) ~ -0.2). If I wanted to calculate the magnitude of the electric field vector at some radius r, how would I do this?

Let's say I wrap up a portion of the charge distribution in a Gaussian sphere of radius r. The net flux through all tiles on this spherical Gaussian surface would be 4*pi*(r^2)*E, where E is the magnitude of the electric field at all points on the surface. Then, using Gauss's law and solving for E, I get

E = [k / (r^2)]Qenc(r)

Where Qenc(r) is the total charge enclosed by the closed surface at radius r. This is where I get stuck. Knowing the charge density function p(r) (in units C / m^3, let's say), how do I find the net enclosed charge by the sphere of radius r?

Any help would be appreciated.

 

[nrqed]

I understand how to use Gauss's law to calculate the electric field at some point for say a sphere with charge distributed uniformly in it. I am bit confused, though, about calculating the electric field at some point for a non-uniform charge distribution.

For example, say that I have a spherically symmetric negative charge distribution around the origin of my reference frame, with charge density p(r) given by -0.5e^(-r), where r is the distance from the origin (IE, the radius of a given sphere) (so, p(0) = -0.5 and p(1) ~ -0.2). If I wanted to calculate the magnitude of the electric field vector at some radius r, how would I do this?

Let's say I wrap up a portion of the charge distribution in a Gaussian sphere of radius r. The net flux through all tiles on this spherical Gaussian surface would be 4*pi*(r^2)*E, where E is the magnitude of the electric field at all points on the surface. Then, using Gauss's law and solving for E, I get

E = [k / (r^2)]Qenc(r)

Where Qenc(r) is the total charge enclosed by the closed surface at radius r. This is where I get stuck. Knowing the charge density function p(r) (in units C / m^3, let's say), how do I find the net enclosed charge by the sphere of radius r?

Any help would be appreciated.

By definition, the charge in a volume V is the integral of the volume charge density p over that volume, q_{enc}= \int dV \rho. In your case, it is obviously easier to work in spherical coordinates and since the charge density is independent of the angle, you simply get q_{enc}= 4 \pi \int_0^R dr r^2 \rho(r) where R is the location of the point where you want to evaluate the E field.

(I am assuming that this point is inside the charged sphere. If the point is
outside of the sphere then you obviously integrate up to the radius of the sphere only)

Patrick

 

[durt]

You'll have to integrate. Imagine thin spherical shells a distance r from the origin of thickness dr.

 

[Signifier]

Okay, so I have a spherical shell of radius r and thickness dr. The volume enclosed by this shell would be the surface area of the sphere multiplied by the thickness (?), or 4*pi*(r^2)*dr. The charge enclosed by this would then be 4*pi*(r^2)*p(r)*dr. Then, to get the net charge enclosed by all of these shells from radius r = 0 to radius r = R, I integrate this expression from 0 to R as nrqed has.

Thanks durt and nrqed, I get it now!