91Ó°ÊÓ

The electric field is a fundamental concept in electromagnetism, representing the force per unit charge that a charge would experience at a point in space. In our given problem, the electric field is expressed in spherical coordinates as \( \mathbf{E}=k r^{3} \hat{\mathbf{r}} \), where \( k \) is a constant and \( r \) represents the radial distance from the origin.
This form of the electric field signifies that the magnitude of the field increases with the cube of the distance from the origin. It radiates outward symmetrically, which is typical in spherical coordinates. This setup is convenient for problems with spherical symmetry, as it simplifies calculations and helps us understand how electric fields behave in complex, three-dimensional spaces.

The fact that the electric field solely points in the radial direction \( \hat{\mathbf{r}} \) implies that it affects charges radially outward or inward, depending on the sign of \( k \). This radial dependence is crucial for applying Gauss's law in later sections, as it ensures symmetry simplifying the integration process.

Charge density, denoted as \( \rho \), quantifies the amount of charge per unit volume at a point in space. In this exercise, we determine the charge density via Gauss's law, which connects electric fields and charge distributions.
According to Gauss's law in its differential form, \( abla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \), where \( \varepsilon_0 \) is the permittivity of free space, and \( abla \cdot \mathbf{E} \) represents the divergence of the electric field.
We calculate this divergence for the given electric field \( \mathbf{E}=k r^{3} \hat{\mathbf{r}} \) and find:

• Divergence: \( abla \cdot \mathbf{E} = 5kr^2 \)
• Therefore, \( \rho = \varepsilon_0 5kr^2 \)

This result illustrates that the charge density increases with \( r^2 \). It means the farther you go from the origin, the more densely packed the charge becomes. Such a relationship aligns with the intuition that an increasing electric field with distance would require more charge at further distances.

Spherical coordinates are a system for representing points in three-dimensional space, particularly useful for problems possessing radial symmetry. These coordinates consist of three parameters: radial distance \( r \), polar angle \( \theta \), and azimuthal angle \( \phi \).
The radial distance \( r \) measures the distance from the origin to the point, the polar angle \( \theta \) is measured from the positive z-axis, and the azimuthal angle \( \phi \) is the angle in the xy-plane from the positive x-axis.
In our exercise, spherical coordinates simplify the mathematical handling of the electric field \( \mathbf{E}=k r^{3} \hat{\mathbf{r}} \). They allow for more straightforward integration and application of Gauss's law, particularly beneficial for calculating total charge in spherical shells.

The integrals to find total charge employ these coordinates, optimizing the computation of volumes and surfaces in a sphere. The symmetry inherent in spherical coordinates is why they are preferred for spherical fields like ours, where everything changes consistently with the radius "r" rather than in "cartesian" directions.

Finding the total charge within a spherical region involves integrating the charge density over the sphere's volume. This process is done here in two distinct methods, both illustrating the power of Gauss's law.
The first method involves directly integrating the charge density \( \rho = \varepsilon_0 5kr^2 \) over the defined spherical volume:

• The integral formulates as \( Q = \varepsilon_0 5k \int_0^R \int_0^{\pi} \int_0^{2\pi} r^4 \sin\theta \, d\phi \, d\theta \, dr \)
• Resulting in \( Q = 2 \pi \varepsilon_0 k R^5 \)

The second method harnesses the integral form of Gauss's Law. It equates the electric flux through a spherical surface with the enclosed charge:

• Flux \( \Phi = \int_0^R \int_0^{\pi} \int_0^{2\pi} k r^3 \cdot r^2 \sin \theta \, d\phi \, d\theta \, dr = 4 \pi k R^5 \)
• Charge from Gauss's Law: \( Q = \varepsilon_0 \Phi = 4 \pi \varepsilon_0 k R^5 \)

Both approaches give a nuanced understanding and validate each other's results. Observing any discrepancies might suggest calculation errors, thus ensuring both conceptual understanding and mathematical rigor.