A copper sphere of radius \(4 \mathrm{~cm}\) carries a uniformly distributed total charge of \(5 \mu \mathrm{C}\) in free space. \((a)\) Use Gauss's law to find \(\mathbf{D}\) external to the sphere. (b) Calculate the total energy stored in the electrostatic field. (c) Use \(W_{E}=\) \(Q^{2} /(2 C)\) to calculate the capacitance of the isolated sphere.

The electric displacement vector, often denoted as \( \mathbf{D} \), is a vector field that appears in Maxwell's equations. It accounts for the effect of free charge within materials in creating an electric field. For an intuitive grasp, consider it as a measure of the electric field strength that accounts for the material's properties in which it exists. In a vacuum or free space, the electric displacement vector is directly proportional to the electric field \( \mathbf{E} \) and is given by the equation \( \mathbf{D} = \epsilon_0 \mathbf{E} \), where \( \epsilon_{0} \) represents the vacuum permittivity.

When using Gauss's Law, one of the fundamental equations in electrostatics, to calculate the electric field around a charged object, we can also determine the electric displacement vector. In the context of our exercise, the sphere's symmetric charge distribution simplifies the calculation, as the electric field at any point outside the sphere only depends on the total charge and the distance from the center of the sphere. The law states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by \( \epsilon_0 \). Thus, \( \oint_{S}\mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\text{enc}}}{\epsilon_{0}} \), where \( Q_{\text{enc}}\) is the enclosed charge.

By considering a Gaussian surface (a convenient mathematical construct) outside of the sphere and knowing the charge distribution, we can solve for the electric displacement vector \( \mathbf{D} \) at any point outside the sphere.

The electrostatic field energy is the energy stored in the electric field of a system of charged particles. To understand this, imagine that assembling the charges involves work, which then becomes stored as potential energy in the system. The total energy stored in an electrostatic field \( W_E \) can be expressed through the volume integral of the energy density of the electric field, which is given by the formula \( W_E = \frac{1}{2}\epsilon_0 \mathbf{E}^2 \).

In our exercise, the electrostatic field energy is calculated for the region around a charged sphere. By integrating the energy density (\( \epsilon_0 \mathbf{E}^2 \) or \( \mathbf{D} \cdot \mathbf{E} \) in the case of dielectrics) over the volume outside the sphere, we determine the total energy stored in the field. This integration is simplified by the spherical symmetry of the charge distribution, which allows the electric field to be treated as a function of radius alone.

The integral extends from the surface of the sphere to infinity since that's where the field exists. However, the contribution of regions far away from the sphere decreases rapidly because the electric field strength diminishes with distance, following Coulomb's law.

Capacitance is a measure of an object's ability to store charge per unit voltage; it is a fundamental property of conductors. For a sphere in free space, the capacitance can be determined by considering the relationship between the stored energy in the electrostatic field and the charge on the sphere. Mathematically, the capacitance \( C \) of a conductor is defined as the ratio of the charge \( Q \) it holds to the potential \( V \) it develops; in the case of a sphere, the potential is due to the sphere itself.

Mathematically, the energy stored in the electrostatic field of a charged sphere can be related to its capacitance through the formula \( W_E = \frac{Q^2}{2C} \), which is derived from work done in charging the capacitor. By rearranging this equation \( W_E = \frac{Q^2}{2C} \), we find \( C = \frac{Q^2}{2W_E} \). This equation gives us the capacitance directly in terms of the known charge and the calculated electrostatic field energy.

In the example of the copper sphere with a known charge, we can use the energy calculated in the previous step to find the sphere's capacitance. Since the sphere is isolated, there's no external electric potential to consider, making the calculation straightforward. Remember, the calculated capacitance of a sphere implies an isolated condition; in practical scenarios with other conductors and materials nearby, the actual capacitance may be different.