(III) Charge is distributed within a solid sphere of radius...

(III) Charge is distributed within a solid sphere of radius $r_{0}$ in such a way that the charge density is a function of the radial position within the sphere of the form: $\rho_{\mathrm{E}}(r)=\rho_{0}\left(r / r_{0}\right) .$ If the electric field everywhere within the sphere in terms of $Q, r_{0},$ and the radial position $r ?$the total charge within the sphere is $Q$ (and positive), what is the electric field everywhere within the sphere in terms of $Q, r_{0},$ and the radial position $r ?$

Key Concepts

Integration in Spherical Coordinates

Integration in spherical coordinates is the mathematical process used to calculate quantities, such as the total enclosed charge, over a spherical volume. This involves integrating the charge density multiplied by the volume element expressed in spherical coordinates. The use of spherical coordinates is particularly beneficial in problems with spherical symmetry, as it aligns with the natural geometry of the problem and simplifies the integration process.

Charge Density

Charge density is a measure of the amount of electric charge per unit volume at a given point in space. When the charge density is not uniform but varies with position, it must be integrated over the volume to determine the total charge contained within a region. In the context of a sphere with radially dependent charge density, this concept allows one to calculate the charge enclosed by a Gaussian surface of any radius inside the sphere.

Electric Field Distribution in a Continuous Medium

The electric field distribution in a medium with a continuous charge distribution is determined by the cumulative effect of all the charges present. When the charge distribution is defined by a function of the radial position, the electric field inside the sphere is obtained by first calculating the enclosed charge through integration and then applying Gauss's Law. This approach provides a systematic way to relate the spatial variation of the charge density to the electric field at different points within the sphere.

Gauss's Law

Gauss's Law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the total charge enclosed by that surface. It is mathematically expressed as the surface integral of the electric field over a closed surface being equal to the enclosed charge divided by the permittivity of free space. This law is particularly useful in solving problems with high symmetry, such as spherically symmetric charge distributions, where it simplifies the calculation of the electric field.

Spherical Symmetry

Spherical symmetry indicates that the physical properties of a system, such as the charge density or the electric field, depend only on the radial distance from the center and not on the direction. This symmetry is crucial because it allows one to choose a spherical Gaussian surface, making the application of Gauss's Law straightforward. In such systems, the electric field is radial and has the same magnitude at every point equidistant from the center.

Transcript

00:01 Okay, so we're doing chapter 22, problem 49 here.

00:04 And this says charge is distributed within a solid sphere of radius r -not in such a way that the charge density is a function of the radial position.

00:16 So it's given as the charge density as a function of r is row not times r over r -not.

00:23 So if the total charge within the sphere is q and is positive, what is the electric field everywhere within the sphere in terms of? q, r0, and r.

00:36 Ok.

00:37 So first we know that the symmetry of the charge distribution allows this to be calculated using a gaussian spherical surface.

00:45 So for a radius of r less than r0, the enclosed charge is going to be the integral of our charge density function times dv.

01:02 So we can rewrite this in terms of just the radial distance because we know that the volume and row only depend on the radial distance.

01:13 So this now integrating from zero to r.

01:16 This is row not times r prime over r0 and dv when we're changing into dr, we pick up the 4 pi r squared, r squared, dr prime.

01:34 So this becomes our integral now to solve for what the enclosed charges and we can actually just solve for this.

01:41 So this becomes p -mat over r -not times 4 pi integral from 0 to r of r -pribue to r prime.

01:55 If we integrate that, we get row -not pi -r to the fourth over r -not.

02:02 So this is our charge enclosed as a function of r...

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