Suppose the electric field in some region is found to be 𝐄=k r^3 𝐫̂, in spherical coordinates (

step 1 Hi there, so for this problem, we need to suppose that the electric field in some region is foun

Suppose the electric field in some region is found to be 𝐄=k...

Suppose the electric field in some region is found to be $\mathbf{E}=k r^{3} \hat{\mathbf{r}},$ in spherical coordinates ( $k$ is some constant). (a) Find the charge density $\rho$. (b) Find the total charge contained in a sphere of radius $R$, centered at the origin. (Do it two different ways.)

Suppose the electric field in some region is found to be $\mathbf{E}=k r^{3} \hat{\mathbf{r}},$ in spherical coordinates ( $k$ is some constant).
(a) Find the charge density $\rho$.
(b) Find the total charge contained in a sphere of radius $R$, centered at the origin. (Do it two different ways.)

Key Concepts

Gauss's Law (Differential Form)

This principle relates the divergence of the electric field to the local charge density, stating that the divergence of E is equal to the charge density divided by the permittivity of free space. It forms the basis for finding the charge density distribution when the electric field is given, by setting up the equation ?·E = ?/??.

Divergence in Spherical Coordinates

In problems with spherical symmetry, the divergence operator must be expressed in spherical coordinates. The formula involves derivatives with respect to the radial coordinate and angular coordinates, but for radially symmetric fields it simplifies considerably. Understanding how to apply this operator is crucial for converting the electric field into its corresponding charge density.

Volume Integration in Spherical Coordinates

To find the total charge contained within a volume, the charge density is integrated over that volume. In spherical coordinates, this involves including the proper Jacobian (r² sin?) in the integral. This technique ensures that the integration accounts for the geometry of space in spherical symmetry.

Divergence Theorem

This theorem connects the flux of a vector field through a closed surface to the volume integral of its divergence over the region enclosed by the surface. It provides an alternative method for computing the total charge by equating a surface integral of the electric field to the volume integral of the charge density, which is particularly useful in confirming results obtained through direct integration.

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