A solid sphere of radius R carries a nonuniform volume...

A solid sphere of radius $R$ carries a nonuniform volume charge density $\rho=\rho_{0} e^{r / R},$ where $\rho_{0}$ is a constant and $r$ is the distance from the center. Find an expression for the electric field strength at the sphere's surface.

Key Concepts

Integration in Spherical Coordinates

Integration in spherical coordinates is used to compute the total charge enclosed within a spherical volume when the charge density depends on the radial coordinate. By setting up the appropriate volume element in spherical coordinates and integrating over the radius, angles, and the given charge density function, one can determine the total charge that contributes to the electric field.

Volume Charge Density Distribution

A volume charge density describes how charge is distributed throughout a three?dimensional region. When the charge density is given as a function of the radial distance, it indicates that the charge is not uniformly distributed but varies with distance from the center. Understanding the form of this distribution is crucial for setting up the integration to determine the total enclosed charge.

Spherical Symmetry

Spherical symmetry implies that the physical properties of a charge distribution are the same in every direction from the center. This symmetry allows us to assume that the electric field at any point on a spherical surface (centered at the distribution's center) is radial and has the same magnitude, which greatly simplifies the application of Gauss's Law.

Gauss's Law

Gauss's Law is a fundamental principle in electrostatics stating that the total electric flux passing through a closed surface is proportional to the total charge enclosed by that surface. This principle is especially powerful for systems exhibiting high symmetry, as it simplifies the calculation of the electric field by converting complex integrals into algebraic equations based on symmetry arguments.

Transcript

00:01 Okay, so in this problem, we have a solid sphere, and it has a certain volume charge density, and we want to get the electric field strength at the surface.

00:10 And so first we need to get the total charge, because at the surface, all that matters is the total charge in close.

00:19 You can make a gauces law sort of argument by drawing a galsa sphere at the, at the, sorry, the radius of the sphere.

00:31 And you can see that you're really just interested in the total charge in the sphere, which is the total charge enclosed.

00:41 So let's first get the total charge.

00:43 So we can do q is the integral of row not.

00:47 So we're going to do the integral of row dv, and then that's row not.

00:51 E to the r over r.

00:54 And then our dv element is going to be 4 pi r squared, dr.

01:02 So just to be totally clear, this is our dv.

01:05 Our differential volume element is a spherical shell with a thickness dr.

01:10 And this is our row.

01:11 And so we want to integrate this to get the total charge.

01:15 So take row not to the outside, row not 4 pi.

01:20 And then we're left with e to the r over r and r squared, and we want to do that from 0 to r.

01:25 And i personally do not remember how to do that integral.

01:29 I think it's like an integration by parts.

01:33 So i'm just going to look it up on wolfram alpha, so e to the r over r times r squared, integrate from r equals 0 to r equals r.

01:47 And then i'll copy and paste what i typed into wolfram alpha to help me get that...

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