A sphere of radius R has total charge Q . The volume charge...

A sphere of radius $R$ has total charge $Q .$ The volume charge density $\left(\mathrm{C} / \mathrm{m}^{3}\right)$ within the sphere is $$\rho=\rho_{0}\left(1-\frac{r}{R}\right)$$ This charge density decreases linearly from $\rho_{0}$ at the center to zero at the edge of the sphere. a. Show that $\rho_{0}=3 Q / \pi R^{3}$ Hint: You'll need to do a volume integral. b. Show that the electric field inside the sphere points radially outward with magnitude $$E=\frac{Q r}{4 \pi \epsilon_{0} R^{3}}\left(4-3 \frac{r}{R}\right).$$ c. Show that your result of part b has the expected value at $r=R.$

A sphere of radius $R$ has total charge $Q .$ The volume charge density $\left(\mathrm{C} / \mathrm{m}^{3}\right)$ within the sphere is $$\rho=\rho_{0}\left(1-\frac{r}{R}\right)$$
This charge density decreases linearly from $\rho_{0}$ at the center to zero at the edge of the sphere.
a. Show that $\rho_{0}=3 Q / \pi R^{3}$ Hint: You'll need to do a volume integral.
b. Show that the electric field inside the sphere points radially outward with magnitude $$E=\frac{Q r}{4 \pi \epsilon_{0} R^{3}}\left(4-3 \frac{r}{R}\right).$$
c. Show that your result of part b has the expected value at $r=R.$

Key Concepts

Volume Integration in Spherical Coordinates

This concept involves calculating the total charge inside a region by integrating a specified charge density over the volume of the region. When the charge density varies with the radius in a sphere, using spherical coordinates simplifies the integration process by naturally aligning with the symmetry of the problem, where the volume element is expressed in terms of the radial and angular coordinates.

Gauss's Law in Electrostatics

Gauss's Law states that the electric flux through a closed surface equals the enclosed charge divided by the permittivity of free space. In problems with spherical symmetry, this principle enables one to relate the electric field at a given radius to the charge enclosed within a spherical Gaussian surface, thereby simplifying the determination of the field from a non-uniform charge distribution.

Radial Symmetry

Radial symmetry implies that the properties of the system such as the charge density and the consequent electric field depend only on the distance from the center of the sphere. This symmetry leads to simplifications in both the integration process and the application of Gauss's Law, since it reduces the problem to one dimension – the radial coordinate.

Boundary Conditions for Electric Fields

When determining the electric field in different regions of a system, especially across boundaries such as the surface of a charged sphere, it is important to check that the field expressions match known physical behavior at those boundaries. This often means verifying that the internal field approaches the well-established external field expression at the surface, ensuring consistency and correctness in the derived solution.

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